Numerical Modeling and Experimental Investigation of Large Deformation Under Static and Dynamic Loading

Master’s Thesis in Sustainable Structural Engineering

Numerical modeling and experimental investigation of large deformation under static and dynamic loading

Author: Benjamin Bondsman Supervisors: Björn Johannessona Andreas Linderholtb Assistant: Winston Mmari Examinator: Björn Johannesson Course code: 5BY31E Semester: VT 2021, 30 credits aDepartment of Building Technology bDepartment of Mechanical Engineering Faculty of Technology Numerical modeling and experimental investigation of large deformation under static and dynamic loading

Benjamin Bondsman

Linnaeus University Faculty of Technology Sustainable Structural Engineering master’s programme

2021 Benjamin Bondsman

Numerical modeling and experimental investigation of large deformation under static and dynamic loading

This work concludes the two years (120 ECTS) Master’s programme in Sustainable Structural Engineering at Linnaeus Univer- sity in Växjö, Sweden. The work has been performed during the Spring 2021.

Field of research: Continuum Mechanics

Author: Benjamin Bondsman

2021 Acknowledgement

This work is result of numerous comprehensive and extensive lectures and supervision given by Professor Björn Johanesson, whom i ﬁnd to exemplify the most signiﬁcant quali- ties & characteristics a teacher and mentor should possess. In this spirit, I’d like to express my sincere gratitude to Prof. Björn Johannesson for his unwavering support throughout of the work. I wish to extend my deepest gratitude and appreciation to my supervisor Associate Professor Andreas Linderholt for his relentless support, insightful suggestions, invaluable guidance and being a tremendous mentor. Besides my supervisors, I’d also like to thank PhD student Winston Mmari for technical consultation and Laboratory Tech- nician Mats Almström for his assistance in setting up the laboratory equipment.

Benjamin Bondsman June 7, 2021 Abstract

Small kinematics assumption in classical engineering has been in the center of consideration in structural analysis for decennaries. In the recent years the interest for sustainable and optimized structures, lightweight structures and new materials has grown rapidly as a consequence of desire to archive economical sustainability. These issues involve non-linear constitutive response of materials and can only be accessed on the basis of geometrically and materially non-linear analysis. Numerical simulations have become a conventional tool in modern engineering and have proven accuracy in computation and are on the verge of superseding time consuming and costly experiments. Consequently, this work presents a numerical computational framework for modeling of geometrically non-linear large deformation of isotropic and orthotropic materials under static and dynamic loading. The numerical model is applied on isotropic steel in plane strain and orthotropic wood in 3D under static and dynamic loading. In plane strain Total Lagrangian, Updated Lagrangian, Newmark-β and Energy Conserving Algorithm time- integration methods are compared and evaluated. In 3D, a Total Lagrangian static approach and a Total Lagrangian based dynamic approach with Newmark-β time-integration method is proposed to numerically predict deformation of wood under static and dynamic loading. The numerical model’s accuracy is validated through an experiment where a knot-free pine wood board under large deformation is studied. The results indicate accuracy and capability of the numerical model in predicting static and dynamic behaviour of wood under large deformation. Contrastingly, classical engineering solution proves its inaccuracy and incapability of predicting kinematics of the wood board under studied conditions.

Keywords: Total Lagrangian, Updated Lagrangian, Newmark, Energy Conserving, Al- gorithm, Numerical, Modeling, Large, Deformation, steel, Wood, Pine, Plane Strain, 3D. Numerisk modellering och experimentell undersökning av stora deformationer vid statisk och dynamisk belastning

Benjamin Bondsman

Linnéuniversitetet Fakulteten för teknik Hållbar konstruktionsteknik, masterprogram

2021 Abstrakt

Små kinematikantaganden inom klassisk ingenjörsteknik har varit centralt för kon- struktionslösningar under decennier. Under de senaste åren har intresset för hållbara och optimerade strukturer, lättviktskonstruktioner och nya material ökat kraftigt till följd av önskan att uppnå ekonomisk hållbarhet. Dessa nya konstruktionslösningar involverar icke- linjär konstitutiv respons hos material och kan endast studeras baserad på geometriskt och materiellt olinjär analys. Numeriska simuleringar har blivit ett konventionellt verktyg inom modern ingenjörsteknik och visat sig ge noggrannhet i beräkning och kan på sikt ersätta tidskrävande och kostsamma experiment. Detta examensarbete presenterar ett numeriskt beräkningsramverk för modellering av geometrisk olinjäritet med stora deformationer hos isotropa och ortotropa material vid statisk och dynamisk belastning. Den numeriska modellen appliceras på isotropiskt stål i plantöjning och ortotropisk trä i 3D vid statisk och dynamisk belastning. I fallet med plantöjning jämförs och utvärderas den Totala Lagrangianen, Uppdaterade Lagrangianen, Newmark-β och Energi Konserverings Algoritm metoderna. I 3D föreslås en statisk Total Lagrangian metod och en dynamisk Total Lagrangian-baserad metod med Newmark-β tidsintegreringsmetod för att numeriskt förutse statisk och dynamisk deformation hos trä. Den numeriska modellens noggrannhet valideras genom ett experiment där en kvist- fri furuplanka studeras under stora deformationer. Resultaten bekräftar noggrannhet och förmåga hos den numeriska modellen att förutse statiska och dynamiska processer hos trä vid stora deformationer. Däremot, visar klassisk ingenjörslösning brist på förmåga att förutse trä plankans kinematik under studerade förhållanden.

Keywords: Total Lagrangian, Updated Lagrangian, Newmark, Energi, Konservering, Algoritm, Numerisk, Modellering, Stora, Deformationer, Stål, Trä, Furu, Plantöjning, 3D. List of Figures

Figure 1 – Narrow Bridge collapse in 1940 [3]...... 12 Figure 2 – Collapse of (a) Maskinhallen and (b) Siemens Arena [16],[10]...... 13 Figure 3 – Geometry & mechanical description of a steel cantilever in 2D...... 15 Figure 4 – Geometry & mechanical description of wood board in 3D...... 15 Figure 5 – Illustration of motion of a continuum material body...... 19 Figure 6 – Illustration of internal material response in section plane: (a) Material body under external loads; (b) Resulting internal material forces; (c) Internal forces in static equilibrium...... 25 Figure 7 – Linearization...... 34 Figure 8 – Illustration of Lagrangian mesh...... 36 Figure 9 – Illustration of Lagrangian (L) and Eulerian (E) mesh...... 37 Figure 10 – (a) Undamped and (b) damped material response under dynamic load. 40 Figure 11 – Stability diagram of Newmark-β Method...... 45 Figure 12 – Stress-strain curve...... 51 Figure 13 – Illustration of anisotropy and orthotropy...... 52 Figure 14 – Orthotropic directions...... 52 Figure 15 – Coordinate transformation of annual rings in wood...... 53 Figure 16 – Illustration of mathematical coordinate transformation...... 53 Figure 17 – Element coordinate transformation ...... 56 Figure 18 – Coordinate transformation of annual rings in wood in 2D...... 57 Figure 19 – Illustration of annual rings pith variation along length of a wood board. 58 Figure 20 – Geometry and mathematical description of coordinate transformation of wood in 3D...... 59 Figure 21 – Illustration of (a) triangular mesh and (b) tetrahedral mesh...... 70 Figure 22 – Illustration of 2D FEM triangular element...... 71 Figure 23 – Illustration of 3D FEM tetrahedral element...... 72 Figure 24 – (a) Analytical integration and (b) numerical integration...... 93 Figure 25 – Schematic illustration of Newton-Raphson scheme...... 96 Figure 26 – Illustration of cantilever geometry in 2D...... 99 Figure 27 – FEM model of steel cantilever in 2D with triangular mesh elements. . . 99 Figure 28 – Illustration of experimental model of the wood board...... 100 Figure 29 – FEM model of the wood board with tetrahedral mesh elements. . . . . 101 Figure 30 – Experimental setup of mechanical clamp...... 106 Figure 31 – Experiment design strategy...... 107 Figure 32 – Non-linear static deformation in Total Lagrangian (TL) and Updated Lagrangian (UL) with colormap of axial stresses in [MPa]...... 109 Figure 33 – Linear static deformation with colormap of axial stresses in [MPa]. . . 109 Figure 34 – Comparison of horizontal displacement inTL,UL vs. linear solution. . 110 Figure 35 – (a) horizontal and (b) vertical static reaction forces in plane strain. . . 110 Figure 36 – (a) horizontal and (b) vertical static internal forces in plane strain. . . 111 Figure 37 – (a) axial (b) transversal and (c) shear strains in plane strain (static). . 111 Figure 38 – (a) axial (b) transversal and (c) shear stresses in plane strain (static). . 112 Figure 39 – Dynamic motion ofTL andUL solution in Newmark- β scheme with colormap of axial stresses in [MPa]...... 113 Figure 40 – Dynamic motion of linear engineering solution in Newmark-β scheme with colormap of axial stresses in [MPa]...... 113 Figure 41 – (a) Vertical (b) horizontal displacement of Linear and non-linear Newmark- β method vs. non-linear Energy Conserving Algorithm...... 114 Figure 42 – (a) horizontal and (b) vertical dynamic displacement in plane strain. . 115 Figure 43 – (a) horizontal and (b) vertical velocity response in plane strain. . . . . 115 Figure 44 – (a) horizontal and (b) vertical acceleration response in plane strain. . . 116 Figure 45 – (a) horizontal and (b) vertical dynamic internal forces in plane strain. . 116 Figure 46 – (a) axial (b) transversal and (c) shear strains in plane strain (dynamic). 117 Figure 47 – (a) Axial (b) transversal and (c) shear stresses in plane strain (dynamic).118 Figure 48 – Static deformation in of wood in Total Lagrangian with colormap of displacements...... 120 Figure 49 – Static deformation inTL versus experimental data with 10% bars. . . . 120 Figure 50 – Horizontal, vertical and lateral displacement of wood...... 121 Figure 51 – Dynamic motion of wood in the proposed numerical model...... 122 Figure 52 – Comparison of experimental data from accelerometer 1 and the corresponding numerical model...... 123 Figure 53 – Comparison of experimental data from accelerometer 2 and the corresponding numerical model...... 123 Figure 54 – Comparison of experimental data from accelerometer 3 and the corresponding numerical model...... 124 Figure 55 – Static (a) Horizontal, (b) vertical and (c) lateral internal and reaction forces and corresponding displacements of wood...... 134 Figure 56 – Strains in wood under static loading in 3D...... 135 Figure 57 – Stresses in wood under static loading in 3D...... 135 Figure 58 – Dynamic (a) Horizontal, (b) vertical and (c) lateral internal forces and corresponding displacements of wood...... 136 Figure 59 – Strains in wood under dynamic loading...... 136 Figure 60 – Stresses in wood under dynamic loading...... 137 List of Tables

Table 1 – Triangle Gauss points...... 94 Table 2 – Tetrahedral Gauss points...... 94 Table 3 – Geometry and material properties of steel...... 99 4 Table 4 – Material properties of pine (Eij and Gij in 10 [MPa]) ...... 100 Table 5 – Pine board’s geometrical data ...... 100 Table 6 – Modal eigenfrequency of wood...... 122 List of Algorithms

1 Numerical integration...... 95 2 Newton-Raphson scheme...... 97 3 Total & Updated Lagrangian in Newton-Raphson...... 103 4 Newmark-β time-integration in Newton-Raphson...... 104 5 Energy Conserving Algorithm in Newton-Raphson...... 105 Acronyms list

TL Total Lagrangian

UL Updated Lagrangian

FEM Finite Element Method

PDE Partial Diﬀerential Equation

ODE Ordinary Diﬀerential Equation

MDOF Multi Degree of Freedom Contents

1 INTRODUCTION...... 11 1.1 Background & context ...... 12 1.2 Aim and purpose ...... 16 1.3 Literature review ...... 16

2 FINITE STRAIN THEORY ...... 18 2.1 Kinematics of deformation ...... 19 2.2 Measure of strain ...... 21 2.2.1 Green-Lagrange strain...... 21 2.2.2 Almansi strain...... 22 2.3 Measure of stress ...... 25 2.3.1 Conservation laws of thermodynamics...... 26 2.4 Principle of virtual work ...... 31 2.4.1 Internal virtual work ...... 32 2.5 Linearization ...... 34 2.5.1 Linearization of virtual strains...... 35 2.5.2 Linearization of virtual work...... 35 2.6 Static formulations ...... 36 2.6.1 Total Lagrangian Formulation ...... 36 2.6.2 Updated Lagrangian Formulation ...... 37 2.7 Dynamic formulations ...... 40 2.7.1 Newmark-β method...... 42 2.7.2 Energy conserving algorithm...... 46 2.8 Constitutive material behaviour ...... 51 2.8.1 Orthotropic coordinate transformation...... 53 2.8.2 Invariants of tensors...... 61 2.8.3 Constitutive model behaviour...... 62 3 FEM FORMULATION ...... 65 3.1 Principle of virtual work ...... 66 3.1.1 Matrix formulation in 2D...... 67 3.1.2 Matrix formulation in 3D...... 68 3.2 Linear Shape functions ...... 70 3.2.1 Triangular element...... 71 3.2.2 Tetrahedral Element...... 72 3.3 Strain displacement ...... 74 3.3.1 Matrix formulation in 2D...... 76 3.3.2 Matrix formulation in 3D...... 77 3.4 Static incremental solution ...... 79 3.4.1 Total Lagrangian solution...... 80 3.4.2 Updated Lagrangian solution...... 83 3.5 Implicit dynamic incremental solution ...... 86 3.5.1 Newmark-β method...... 87 3.5.2 Energy conserving algorithm...... 89 3.6 Numerical integration ...... 93 3.7 Numerical solution technique ...... 96

4 IMPLEMENTATION ...... 98 4.1 Geometry and material properties ...... 99 4.1.1 Steel cantilever in plane strain ...... 99 4.1.2 Pine wood board cantilever in 3D ...... 100 4.2 Numerical implementation ...... 102 4.3 Experimental implementation ...... 106 4.4 Analog ﬁlter design and analysis ...... 107

5 NUMERICAL RESULTS IN PLANE STRAIN ...... 108 5.1 Static results in 2D ...... 109 5.2 Implicit dynamic results in plane strain ...... 113

6 NUMERICAL & EXPERIMENTAL RESULTS IN 3D . . . . 119 6.1 Static results ...... 120 6.2 Implicit dynamic results ...... 122

7 ANALYSIS & CONCLUSION...... 125 7.1 Analysis ...... 125 7.2 Conclusion ...... 127

BIBLIOGRAPHY ...... 128 APPENDIX 130

APPENDIX A – NUMERICAL FUNCTIONS ...... 131 A.1 Green-Lagrange strain and deformation gradient ...... 131 A.2 Internal forces ...... 132 A.3 Tangent stiﬀness matrix ...... 133

APPENDIX B – ADDITIONAL NUMERICAL RESULTS IN 3D 134 B.1 Static results ...... 134 B.2 Dynamic results ...... 136 11

Chapter 1

Introduction

n classical engineering, small kinematic assumptions are frequently considered in evaluation of deformation prediction of structures. The most common analysis con- I cerns a linear analysis where deformation is assumed to be small. Over the recent years, the interest in optimized and slender structures, new materials and addressing safety issues of structures has raised exponentially due to global and systematic envi- ronmental challenges and desire to archive economical sustainability. These issues may include collapse or buckling analysis of tall transmission towers, wind turbines, offshore structures or multi-storey timber buildings under severe motion-induced wind load. On safety issues, it may involve progressive damage of nuclear reactor components due to long lasting and high temperature loads to prevent release of radioactive substances to the environment. These issues involve non-linear constitutive response of materials and can only be accessed on the basis of geometrically and material non-linear analysis. There- fore, classical small deformation kinematic assumptions are subjected to limitations. Physical laws of space and time-dependent problems are expressed in terms of Partial Dif- ferential Equation (PDE):s, which analytical and empirical methods for a vast majority of complicated geometries and problems fail to solve. Contrastingly, numerical tools are demonstrably proven to be efficient and accurate in computation, and are on the verge of superseding costly and time-consuming experimenters. As transitioning to a sustainable world in accordance to UN’s climate goals becomes vital, the need for developing new materials bounded by safety standards and quantitative tools to exploit them efficiently becomes crucial. Consequently, this study is intended to present a computational framework for numerical modeling of geometrically non linear large deformation of isotropic and orthotropic materials under static and dynamic loading. The first phase of this study concerns numerical modeling of geometrically non-linear isotropic materials and application of the model in plane strain. In the second phase, a numerical model in 3D is proposed and applied on orthotropic wood under static and dynamic deformation. The results are compared with corresponding experimental investigation and reasonable results are archived. Chapter 1. Introduction 12

1.1 Background & context

In classical ﬁnite element formulation, small kinematic assumptions are frequently considered, where deformation is assumed to be small. Considering small deformation kinematics corresponding to linear analysis, the following equations are derived.

Ka = f . Static formulation (1) Ma¨ + Ca˙ + Ka = f(t) . Dynamic formulation where the stiffness K and the displacement a are a linear function of the external applied load f. In Dynamic analysis the left hand-side of the equation of motion is expressed in terms of material mass M, nodal acceleration a¨, material damping C, velocity a˙ in addition to material stiffness and nodal displacement. On the right hand-side the external load f(t) is a function of time. The small kinematic assumptions are entered into the equation of motion, through integration over the original volume of the finite elements and allowing the strain displacement matrix to be constant and independent of element displacement [12]. When a structure undergoes severe loading conditions, the structural materials experience large strains, displacements, rotation and non-linear material response. The material fibers experience rotation and stretch, which results in variation in density and internal energy. Consequently, the strain displacement matrix becomes dependent on the element displacements and in predictive modeling of Finite Element Method (FEM) based solutions, all kinematic nonlinearities must be accounted for. The 1940 Tacoma Narrows Bridge in the U.S. collapsed about four month after it’s open- ing, due to wind load of magnitude 64 [km/h]. The bridge’s deck collapsed due to oscillation in an alternating twisting motion, which resulted in severe deformation and non-linear dynamic response.

Figure 1 – Narrow Bridge collapse in 1940 [3]. Chapter 1. Introduction 13

Maskinhallen in Sweden collapsed after 25 years in service in November 2020. The timber construction failed due to heavy wind-induced load, which resulted in intolerable deformation. Siemens Arena roof in Denmark failed under few centimeter of snow. The arched timber structure collapsed due to insuﬃcient strength of the material in relation to the applied loads. The material strength was only 30% of the required strength in the critical point, which caused unbearable deformations and failure.

(a) (b)

Figure 2 – Collapse of (a) Maskinhallen and (b) Siemens Arena [16],[10].

Flexible riser pipes are commonly used in offshore structures due to their low bending stiffness, which results in small radii of curvature with a high pressure capacity. These types of structures are commonly exposed to loads induced by ocean current, therefore large deformation becomes an important phenomena through their lifespan, [9]. Highly flexible structures (HFS) are demanded by the aerodynamic industry when designing space vehicles to overcome speed limitations and reduce self weight. These type of structures are used by NASA space mission launch vehicles, to satisfy slenderness and lightweight properties. These slender structures are highly likely to undergo large deformations, [20]. Slender beams, columns, and slabs are ubiquitously demanded by the construction industry to satisfy aesthetics of complex structures while meeting safety regulations and standards. Buckling and collapse analysis of slender structures are of great interest in providing long lasting and safe structures. In material production, large deformation phenomena may be of great interest in steel during heat treatment, sawn wood through drying process or rubber-like materials and hyperelastic polymers which are commonly used in constructions. For example the use of lightweight fiber-reinforced polymer in bridge decks, [26].

The examples and events presented above reveals that a special attention must be given to large deformation analysis in the aim of ensuring safety and prevention of structural failure. As regards to large strains, displacements and rotations a continuum mechanics approach must be employed, where time variable shall be used to trace finite deformation step-wise. Finite strain regime is a continuum mechanics based approach based on Lagrangian and Eulerian descriptions, in which time-dependent material properties can be scrutinized through finite deformation. The material mechanics may include response Chapter 1. Introduction 14 to different mechanical loading including rate-dependency or effect of material properties such as isotropy or anisotropy. The use of Lagrangian and Eulerian descriptions stem from their ability to track material points through deformation configurations and handle complicated boundaries [2]. In small deformation analysis, the undeformed material properties are assumed to remain unchanged after deformation. This lies in the assumption that variations in kinematics are infinitesimal. Oppositely, in large deformation analysis the undeformed material properties no longer remain the same after deformation, because kinematics are no longer infinitesimal. It is natural that a Finite Element based approach can neither integrate over an unknown volume, nor determine properties from too many unknowns in a constitutive relation. For example, in a stress-strain constitutive relation, none of the mentioned quantities can be determined unless one of them is known. In the Finite strain regime, reference and spatial frame of points in space are expressed in terms of Lagrangian coordinates X and Eulerian coordinates x. These coordinates specify and label material points in the reference frame and spatial frame i.e. initial configuration and current configuration. Formulations in terms of Lagrangian measure of stress and strain where material properties are traced from the reference configuration, and all derivatives and integrals are taken with respect to Lagrangian (material) coordinates X, are designated as Total Lagrangian Formulation (TL). Contrastingly, formulation in terms of Eulerian measures of stress and strain are known as Updated Lagrangian Formu- lation (UL), where derivatives and integrals are taken with respect to Eulerian (spatial) coordinates x. In the aim of determining an effective and a relevant numerical approach,TL andUL based methods are compared in the first phase of this study. The mentioned formulations are recognized as static formulations where the material deformation does not depend on time. When the material deformation depends on time, dynamic impact must be recognized. Hence, in dynamic analysis Newmark-β and an Energy Conserving Algorithm time-integration approaches will be employed and the results will be compared. The numerical model will be calibrated with isotropic steel in plane strain and the results will also be compared with geometrically linear (Engineering) solution. The mentioned com- parisons will be conducted with the aim of determining appropriate methods to build up a numerical model, capable of predicting dynamic behaviour of orthotropic wood under static and dynamic loading.

In the ﬁrst phase, a geometry of cantilever bar with an applied displacement is chosen. Due to the nature of the problem a plane strain assumption will be considered. The geometry and its associated dimensions are shown in Figure3. Chapter 1. Introduction 15

y 0[mm] 20 0 [mm] 200 x

500 [mm]

Figure 3 – Geometry & mechanical description of a steel cantilever in 2D.

In the second phase of this study, a cantilever pine wood board is chosen, ﬁxed at one end and on the opposite end a large deformation is applied. The built up numerical model will be examined and validated with an experimental investigation. The geometry and its associated dimensions are shown below. y

Mechanical clamp

z 200 [mm] x

1300 [mm] [mm] 8 0 [mm] 200

~40 [mm]

Figure 4 – Geometry & mechanical description of wood board in 3D.

The imposed displacement will be applied in static analysis and will be utilized as an initial condition in the dynamic analysis. Chapter 1. Introduction 16

1.2 Aim and purpose

This study aims at presenting a numerical computional framework capable of predicting static and dynamic behaviour of orthotropic wood under large deformation where geometrical and material nonlinearities are considered. The study takes into account a comparison of Total Lagrangian and Updated Lagrangian, i.e.TL andUL, as well as Newmark-β and Energy Conserving Algorithm methods with respect to computation- speed and accuracy differences. The results will also be compared with classical small deformation theory. The purpose of this study is to establish a computational numerical model calibrated with experimental investigation and capable of predicting large deformation of orthotropic wood under static and dynamic loading. The given model is intended to provide accuracy and flexibility in generalization and application on different solid continua.

1.3 Literature review

Numerical simulations have become a conventional tool within engineering fields rang- ing from civil engineering, mechanical engineering and mining engineering to aeronautical engineering. These tools have demonstrated accuracy in computation of problems that are too complex to be solved analytically and are on the verge of superseding costly and time-consuming experimenters. In a study of large displacement analysis of a 3D beam [13], Bathe ascertains thatTL and UL yield similar stiffness matrices and nodal force vectors. However, computational effectiveness of each formulation differs. The study highlights the importance of employing appropriate stress and strain measures in development of geometrically non-linear analysis. Bathe’s study confirms that with an increased number of elements, theTL andUL approaches yield matching results. Although the study presents convergence at similar displacements consideringTL with numerous more elements thanUL, the computational speed has not been given attention. In contrast, Belytschko, Liu and Moran [2] state that despite superficial differences in formulation ofTL andUL, the formulations are identical and expressions can be transformed fromTL toUL and vice-versa. This disagreement entails uncertainty whetherTL andUL yield actually identical results despite the differences Lagrangian and Eulerian descriptions inTL andUL. Large deformation tests andTL analysis of flexible beams by Pai, Anderson and Wheater [20], present experimental as well as numerical investigation studying Highly Flexible Struchtes (HFS). Experimental investigation encompasses straight and curved beams, with respect to large bending and twisting. Whereas, numerical analysis comprisesTL formulation using fully and truncated non-linear strain displacement measures and a Green- Lagrange strain. The study presents additionally anUL formulation using truncated Chapter 1. Introduction 17 non-linear strain displacement, Von-Karmen strain. Results corroborates that numerical analysis approaches experimental analysis closely. Non-linear displacement-based FEM analysis of shells by Perngjin and Palazotto [21], presents a modifiedTL formulation for analysis of composite shells undergoing large displacement, rotation and strain-motion. The study investigates composite shells as a consequence of expanded use of high-performance composite materials in mechanical engineering fields, spanning from aerospace vehicles, large space structures, automotive, shipbuilding and recreational industries. The investigation encompasses predictions of static and dynamic response and failure characteristics of composite shells. Authors cor- roborate effectiveness and flexibility ofTL and FEM with accuracy in prediction of large displacement, rotation and strain-motion. Total Lagrangian explicit dynamics has been utilized in Biomedical engineering for analysis of deformation of soft tissues. ATL based numerical algorithm is proposed by Miller, Joldes, Lance and Wittek [11] for analysis of brain and liver tissues with application to surgical simulation. The simulation comprises large deformation analysis of an ellipsoid as an approximation of indentation of brain. Results obtained from theTL are compared with simulation in the commercial software Abaqus. The comparison shows almost identical results, which confirms potential of TL in catching up with experimental results. However, the study does not consider an application ofUL. In study of comparison between classical Newmark-β time integration method and modern Bathe’s ρ∞ method, Noh and Bathe [8] discovered that under certain conditions with respect to stability parameters, Newmark-β and Bathe’s ρ∞ time-integration methods become identical. The stability parameters comprise Average Acceleration Method, originally proposed by Newmark [17]. Energy conservation in Newmark based time integration algorithms by Krenk [14], estab- lishes energy balance equations to examine energy conservation in Newmark. The author ascertains that Newmark time integration with the stability parameters originally proposed by Newmark [17], satisfies energy balance equations and the algorithm is energy conserving. In this work, the material behaviour is considered in accordance to elastic St. Venant- Kirchhoff material model. However, thin incompressible materials such as rubbers and soft biological tissues may under certain conditions be viscoelastic, for which a framework under plane stress condition is proposed by Kroon [15]. Besides of elastic behaviour, materials such as wood at microscopic scale has shown elastic-plastic behaviour due to anatomical features of the material, such as density and proportion of late and early wood and loading directions [1]. The irreversible plastic behaviour may be of significant importance in analysis of heavily loaded wood based structures. Hence, this work can be further developed to scrutinize plastic as well as fracture mechanics of wood as a material and also a member of a structural system. 18

Chapter 2

Finite strain theory

ontinuum mechanics is a branch of mechanics, in which materials are considered as continuous ignoring their true atomic structure. In deriving governing C equation of motion describing material behaviour and constitutive relations, continuum mechanics relies on fundamental physical laws of mass, momentum and energy conservation of thermodynamics. As a branch of continuum mechanics, Finite Strain Theory deals with geometrically non-linear analysis, in which material properties no longer remain unchanged after deformation. Lagrangian (material) coordinates X and Eulerian (spatial) coordinates x are utilized in labeling material points in reference and spatial configuration. In Finite the Strain regime, Total Lagrangian (TL) and Updated Lagrangian (UL) are central, whereTL traces material deformation from initial configurations in which all integrals are derivatives are taken with respect to Lagrangian coordinates X. In anUL approach, material deformation are traced from a previous time-step following material deformation, where all integrals and derivatives are taken with respect to Eule- rian coordinates x. The weak formulation of equations of motion inTL andUL will be developed in this chapter, based on conservation laws of thermodynamics. The discrete equations for finite element approximations associated with each formulation will further be derived. In static analysis with rate-independency, the discrete equations of motion are independent of time. Whereas in dynamics formulation involving rate-dependency, the equations of motion are time-dependent. Hence, aTL approach and an unconditionally stable Newmark- β and Energy Conserving Algorithm time-integration methods are proposed. Considering numerical modeling of orthotropic wood, this chapter conclusively addresses the coordinate transformation technique with respect to annual rings as well as variation of the pith along longitudinal axis and proposes a compatible constitutive material model. Chapter 2. Finite strain theory 19

2.1 Kinematics of deformation

Large deformation kinematics require a consistent continuum mechanics based approach to describe coordinates of particles in a material body undergoing motion and develop the governing equation of motion. In non-linear analysis, it is important to establish equilibrium in the initial conﬁguration of the material body and it is necessary to employ an incremental ﬁnite element approach, where time variable is used to track the loading and motion of the body. The material body is stationed in a Cartesian coordinate system, where it begins to experience large displacement, rotation, strain and non linear constitutive response. The aim is to evaluate equilibrium positions of the body at each discrete time step, where time starts from zero when the body is at rest, and continues incrementally with time-increment ∆t as the body undergoes motion.

Consider a body as in Figure5 being subjected to large displacement denoted by the vector w, which comprises variation in shape and size of the section as the body undergoes large deformation. In order to quantify the variation associated with rate of change in the body, two arbitrary points on the body are chosen. The distance between them AB is denoted by vector dX in the reference configuration and dx in the current configuration at time ∆t. As the body deforms with time, coordinates of the chosen points will also change such that, x = X + w. Hence, two vectors are employed to track the coordinates of each point. Vector X tracks coordinates of point A in the initial configuration, and vector x tracks the new coordinate of the same point in the current configuration. In order to establish a mapping function to link the deformation of the body in the reference configuration to the current configuration, a deformation gradient F is employed such that ∂x ∂(X + w) ∂w dx = F dX ←→ F = = = I + (2) ∂X ∂x ∂X where I is an identity matrix.

x2 χ t = 0 t = ∆t

w A’ A dX x dx B B’ X F

Figure 5 – Illustration of motion of a continuum material body. Chapter 2. Finite strain theory 20

Motion of the deformation gradient F through finite deformation is denoted by χ. The deformation gradient F is a key parameter that describes deformation of the material body at each discrete time step. Deformation gradient F is a second order tensor, hence denoted by Fij and works as a linear transformation matrix, that maps the initial configuration to the current configuration as       ∂x1 ∂x1 ∂x1 dx1 dX1    ∂X1 ∂X2 ∂X3       ∂x2 ∂x2 ∂x2    dx2 =   dX2 (3)    ∂X1 ∂X2 ∂X3          ∂x3 ∂x3 ∂x3 dx3 dX3 ∂X1 ∂X2 ∂X3

A material body undergoes motion with a velocity and an acceleration, which can be determined through the ﬁrst and second derivatives of the motion as ∂χ ∂2χ x˙ = and x¨ = (4) ∂t ∂t2 The velocity gradient L can be expressed in terms of the deformation gradient as

∂x˙ i chain X ∂x˙ i ∂Xk −1 Lij = −−−→ ←→ L = FF˙ (5) rule ∂xj k ∂Xk ∂xj | {z } | {z } F˙ F −1

The velocity gradient is not a symmetric matrix, but it can be decomposed to a symmetric part and a skew symmetric part as 1 1 L = (L + LT ) + (L − LT ) ←→ L = D + W (6) 2 2 | {z } | {z } Symmetric D skew symmetric w

A displacement of any arbitrary point on the body from initial to current conﬁguration, is the diﬀerence between its Eulerian (spatial) coordinates x and Lagrangian (material) coordinates X. w = x − X (7)

The rate of change in displacement, as the material body undergoes different configurations, with respect to the initial configuration is expressed in terms of the displacement gradient as ∂wi Hij = ←→ H = F − I (8) ∂Xj Virtual variation in displacement with respect to the current configuration x, is denoted by grad(δw), whereas in initial configuration X, as GRAD(δw). In the aim of evaluating unknown material quantities, a relation between initial and current configuration is desired. In this relation, displacement variation in both configurations and the mapping function F shall be expressed in terms of one another. In Chapter 2. Finite strain theory 21 this manner, the rate of virtual variation in displacement in the current configuration is expressed in terms of the initial configuration as

grad(δw) = GRAD(δw)F −1 (9) where F = GRAD(w) + I (10)

The virtual variation of the deformation gradient becomes

δF = GRAD(δw) + δI = GRAD(δw) (11)

2.2 Measure of strain

In a continuum mechanics based approach, the employed strain measure must return identical magnitude irrespective of orientation of the material body. This criteria is crucial in formulation of small and large strain, large displacement and rotation to meet compatibility criteria. The invariant objectivity of strain is derived in Section 2.8.1. The strain measure shall not be inﬂuenced by rigid body rotations. Hence, the right Cauchy- Green tensor F T F is utilized in formulation of strain measure and an identity matrix of the size of the deformation gradient F is used to represent the rigid body. It must be emphasized that when a material body is rigid and stationary, the deformation gradient returns a unit matrix.

2.2.1 Green-Lagrange strain

A strain measure that obeys the mentioned requirements is the Green-Lagrange strain with property of tracing material properties from reference conﬁguration, where material properties are known. Hence, this strain measure must be compatible with Lagrangian (material) coordinates X. Green-Lagrange strain can be utilized in formulating small as well as large strain analysis, [12].

∆L2 = ds2 − dS2 −→ ∆L2 = dx · dx − dX · dX ∆L2 = (F dX) · (F dX) − dX · dX (12) ∆L2 = dX · (F T F )dX − dX · (I)dX ∆L2 = dX · F T F − I dX where stretch of the material body is quantified in a similar manner as in Equation (2), in terms of the deformation gradient F . In the above Equation, a new term is defined as F T F − I, which is defined as 2E [19], where the Green-Lagrange strain is expressed as

1 1 ∆L2 EG−L = (F T F − I) ←→ EG−L = (13) 2 2 dX · dX Chapter 2. Finite strain theory 22

In the equation above, it can be observed that the strain tensor EG−L is similar to the engineering strain measure, as a quotient of elongation ∆L2 and original length (dXdX) in initial conﬁguration. However, in continuum mechanics and numerical analysis the Green-Lagrange strain measure is considered as more accurate than the engineering strain measure [19]. A comparison of Green-Lagrange and engineering strain is presented further in this work. The Green-Lagrange strain tensor can be expressed in terms of the displacement gradient H as 1 1 EG−L = (F T F − I) = (H + I)T (H + I) − I 2 2 1 = HT H + HT I + IH + I − I (14) 2 1 1 = H + HT + HT H 2 2 The rate of change in the Green-Lagrange strain measure through conﬁgurations can be expressed as symmetric part of the velocity and deformation gradient, in the aim of compatibility as 1 1 E˙ G−L = F˙ T F + F T F˙ 2 2 1 = F T LT F + F T LF (15) 2 1 = F T LT + L F = F T DF 2 where F T LT = F˙ and LF = F˙ , see Equation (5).

2.2.2 Almansi strain

Strain in the current configuration, also called physical strain, may be formulated in terms of Almansi strain EA, which is independent of rigid body rotation fulfilling the criteria introduced previously. However, Almansi strain is often of interest in small strain analysis with large deformations. Variation in material body stretch with respect to current configuration can be derived as

∆L2 = ds2 − dS2 −→ ∆L2 = dx · dx − dX · dX ∆L2 = dx · dx − (F −1dx) · (F −1dx) (16) ∆L2 = dx · (I)dx − dx · F −T F −1 dx ∆L2 = dx · I − F −T F −1 dx

In the above Equation, a new term is deﬁned as I − F −T F −1, which corresponds to 2E. Hence, the Almansi strain can be expressed as 1 EA = I − F −T F −1 (17) 2 where EA is Almansi strain. The equation above reveals that the physical strain EA is an inverse of Green-Lagrange strain EG−L. This type of strain is often referred to as Chapter 2. Finite strain theory 23

Eulerian-Almansi strain tensor in some textbooks. In continuum mechanics of ﬁnite deformation, it is preferable to analyse deformation according to principle of virtual work, Section (2.4). In this regard it is important to construct virtual variations of strain measure. Recalling (13), where Green-Lagrange strain is expressed in terms of the deformation gradient, allows expressing virtual variation in strains as 1 1 1 δEG−L = δ(F T F − I) = (δF T )F + F T δF (18) 2 2 2 where δ has commutative property. In (11), virtual variation of the deformation gradient δF is expressed in terms of virtual displacement i.e. δF = GRAD(δw). This relation can be utilized to express Green-Lagrange strain in terms of displacement as

1 h i 1 δEG−L = GRAD(δw)T F + F T GRAD(δw) (19) 2 2

Corollary 2.2.1 The transpose of a product of two matrices, is the product of the product in reverse order. T AT B = BT A

Utilizing corollary (2.2.1), virtual variation in the Green-Lagrange strain can be expressed as 1 h iT 1 δEG−L = F T GRAD(δw) + F T GRAD(δw) 2 2 (20) = sym F T GRAD(δw) In the equation above the virtual variation in the Green-Lagrange strain δEG−L yields a symmetric matrix. An expression for virtual variation in Almansi strain δEA can be obtained by multiplying virtual variation in Green-Lagrange strain δEG−L by the deformation gradient F . It is important to note that Almansi strain is considered as physical strain in the current conﬁguration, which is unknown. Therefore, it is desired to be evaluated and expressed in terms of known quantities. Multiplying virtual variation in Green-Lagrange strain EG−L from the left and right sides with F −T and F −1 yields 1 1 F −T δEG−LF −1 = F −T δF T FF −1 + F −T F T δFF −1 2 | {z } 2 | {z } I I (21) 1 1 = F −T δF T + δFF −1 2 2

Corollary 2.2.2 If a matrix is invertible, its inverse is unique. A product of a matrix and its inverse yields an identity matrix I. Chapter 2. Finite strain theory 24

AA−1 = I AT A−T = I

From Equation (19) the virtual variation in the Green-Lagrange strain δEG−L is expressed as 1 h iT 1 δEG−L = F T GRAD(δw) + F T GRAD(δw) 2 2 (22) = F T GRAD(δw) Together with Equations (11) and (21) the virtual variation in the Green-Lagrange strain can be expressed accordingly. 1 1 F −T δEG−LF −1 = F −T [GRAD(δw)]T + GRAD(δw)F −1 2 2 (23) 1 1 = [grad(δw)]T + grad(δw) = δEA 2 2 where it is known since Equation (9) that the following relation between gradient of the displacement in the initial and spatial conﬁguration holds.

grad(δw) = GRAD(δw)F −1 (24)

The virtual variation in Almansi strain δEA is symmetric, as virtual variation in Green- Lagrange strain δEG−L. 1 1 δEA = [grad(δw)]T + grad(δw) 2 2 (25) = sym (grad(δw))

It is important to note that in Equation (23), a relationship between virtual variation of strains in the spatial frame, where properties of the material body are unknown, and initial conﬁguration, where properties of the material body are known has been established. This relationship can be used to track virtual variation of strains from reference frame, as the material body undergoes motion.

δEA = F −T δEG−L F −1 (26) | {z } | {z } Spatial frame Ref. frame

In large deformation continuum mechanics, a collection of strain measures are available, such as right and left Cauchy-Green deformation tensor, Logaritmic strain tensor and Seth-Hill Family of strain tensors. However in this work, Green-Lagrange strain inTL formulation and Almansi strain inUL formulation will be utilized. The choice of strain measures is strongly dependent on compatibility with the chosen stress measures. Chapter 2. Finite strain theory 25

2.3 Measure of stress

Deformation of a material body generates stress or force per unit of area, which plays a central role in generating internal forces to excite internal material response. The material response is mathematically formulated by traction vectors, stress tensors and diverse stress measures in state of stress.

Section Plane N

fint M0

Fc V s2 Fc V s1 Qs x Qs Fc Qs

Tx z Tx y (a) (b) (c) Figure 6 – Illustration of internal material response in section plane: (a) Material body under external loads; (b) Resulting internal material forces; (c) Internal forces in static equilibrium.

Figure6(b) presents a material body exposed to a concentrated load Fc and a surface load Qs and torque Tx about the longitudinal axis. These external loads excite internal response in the material i.e. the material resists the external loads to the best of its capacity. Internal response of a material can be visualized through scrutinizing a section plane of the body, Figure6(b). The sectioned view shows a set of internal forces as consequence of excitation by the external loads. Size and orientation of the response f int depends on cumulative eﬀect of the external loads. The description of internal response relies heavily on a complete analysis of all the forces acting on the body. However, in continuum mechanics, according to the principle of virtual work, all external loads acting on a material body is assumed to be in static equilibrium with the internal response. Figure6(c) demonstrates a set of internal forces following application of static equilibrium. These forces are internal quantities generated by the internal response f int, which are used in continuum mechanics to deﬁne magnitude of internal response due to external excitation. Since the internal forces are acting on a surface, it is appropriate to measure intensity of these forces per unit of area. The intensity per area is the measure of internal stress, excited by external forces. Therefore, higher stress over the same unit of area follows higher internal response and vice versa. In order to derive appropriate stress measures, conservation laws of physics must be taken into account. Chapter 2. Finite strain theory 26

2.3.1 Conservation laws of thermodynamics

Conservation laws in continuum mechanics define rate of change in a physical quantity over a domain with respect to time. These laws are introduced in form of balance equations where rate of change is equivalent to internal production and flux. ∂Ψ = Π + Φ (27) ∂t Ψ is the physical quantity over time t, equal to production within the body Π and a boundary flux Φ.

2.3.1.1 Mass balance

In continuum mechanics, mass is considered to not change magnitude through motion. However, density of a material body varies as the material undergoes motion. Density in the initial configuration is known, i.e. a quotient of mass and volume. Hence, based on mass balance a relation can be established in the initial configuration to track density variation through finite deformation. A mass balance for density variation can be established as ∂ρ = −div(x˙ ρ) (28) ∂t where the density ρ varying over time t is equivalent to divergence of change in the density with respect to velocity of the motion x˙ (x, t). It is important to note that the partial derivative operator is a spatial time derivative operator and refers to a fixed spatial frame. The motion is denoted by χ, Figure5. Since the density varies as the material undergoes finite deformation, it can be expressed as a function of Eulerian (spatial) coordinates and time ρ(x, t) in the spatial frame. Recalling transformation from reference to spatial frame in (2) allows expressing the density in the spatial frame as

ρX = ρ detF (29)

2.3.1.2 Linear momentum balance

When a material body undergoes motion, tendency of continuity arises. The momentum is a measure of continuity tendency of a material body under motion to keep moving. According to Euler’s theorem, rate of change in momentum is equivalent to the external applied loads. Since stress is a function of external applied loads per unit of area, the momentum balance for an object undergoing motion can be expressed in terms of stress and body force. ρx¨ = div(T ) + ρb (30) where material derivative of the velocity i.e. acceleration x¨, is tracking rate of change in the velocity of the material points under motion with respect to time. This variation according to (4) results in variation in density. This variation is equivalent to divergence Chapter 2. Finite strain theory 27 of stress T and the body force density ρb, where ρ is mass density and b is body force. It is important to note that the stress is expressed in terms of Cauchy stress deﬁned in the spatial frame. Hence it is dependent on Eulerian coordinate and time T (x, t).

2.3.1.3 Internal energy balance

Internal energy balance in continuum mechanics originate from the ﬁrst law of thermodynamics. The law recognizes that rate of variation in energy of a continuous system is equivalent to internal work done by the system and energy production within the system. The energy balance can be expressed in terms of density as a function of energy variation as ρe˙ = −div(q) + tr(T T L) + ρr (31) where material-derivative of the internal energy and external heat ﬂux are denoted by e˙ and q. The last term is associated with density variation due to radiation r. According to balance of angular momentum, energy balance for non-polar materials is given in terms of Cauchy stress such that T T = T . The velocity gradient in (6) has a symmetric and skew symmetric part. The the skew symmetric part of the velocity gradient does not produce any energy. Therefore, the velocity gradient in the equation above can be directly expressed in terms of its symmetric part D since L = D + W . The symbol "tr" represents trace, which is used in mathematics to sum elements in the main diagonal of a square matrix. Considering the mentioned condition, the energy equation can be expressed as ρe˙ = −div(q) + tr(TD) + ρr (32) where material derivative of internal energy, i.e. e˙, represents variation in internal energy.

2.3.1.4 Entropy inequality

Entropy inequality describes energy production in a material body through deformation. Initially the entropy inequality is utilized in classical thermodynamics, however it is extended to mechanics of continuum system as q ρr ρη˙ ≥ −div + (33) θ θ where time derivative of the entropy and temperature are denoted by η˙ and θ˙. It is important to bear in mind that time-derivative, also called material-derivative, is following the material coordinates through deformation. The entropy according to Clausius–Duhem inequality can be expressed by combining internal energy balance equation and entropy inequality, i.e. (32) and (33). As previously stated, deformation of a body results in energy production, hence, energy is always a positive quantity and can not decrease due to deformation. Therefore temperature θ Chapter 2. Finite strain theory 28 must be consequently positive too. Combining Equations (32) and (33) yield   1 q θρη˙ + θ div −ρr ≥ 0 (34) θ θ | {z } The under-braced term in the equation above can be mathematically rewritten as

θ q! q q div(q) = div = θdiv + grad(θ) (35) θ θ θ

Inserting the divergence of the ﬂux q into Equation (34) yield 1 q θρη˙ + div(q) − grad(θ) − ρr ≥ 0 (36) θ θ Substituting the expression in (32) for density variation due to radiation (ρr) into the equation above, yield the following q θρη˙ + div(q) − grad(θ) − div(q) − ρe˙ + tr(TD) ≥ 0 θ q (37) θρη˙ − grad(θ) − ρe˙ + tr(TD) ≥ 0 θ In order to determine the entropy from its derivative as expressed above, Helmholtz free energy is utilized such that ψ = e − ηθ (38)

Diﬀerentiating Helmholtz free energy equation in the aim of scrutinizing rate of energy variation in the system, yield

e˙ ηθ˙ ψ˙ ψ˙ =e ˙ − ηθ˙ − ηθ˙ ⇐⇒ η˙ = − − (39) θ θ θ The expression above inserted into (37), yield Clausius-Duhem inequality, upon which stress measurement lies. q tr(TD) − ρηθ˙ − grad(θ) − ρψ˙ ≥ 0 (40) θ Considering isothermal assumption, such that temperature is constant, the Clausius- Duhem inequality can be reduced to the following

Isothermal assumption: θ˙ = 0; grad(θ) = 0 (41) tr(TD) − ρψ˙ ≥ 0

In finite deformation, the spatial configuration of the material body at time t + ∆t is unknown, Figure5. This is a fundamental difference compared to linear analysis where kinematics are assumed to be infinitesimal and material properties are assumed to remain unchanged after deformation. In large deformation theory, the properties of a material body i.e. volume, density internal energy and displacements are no longer identical after Chapter 2. Finite strain theory 29 deformation. Cauchy stress T , also called Eulerian stress, is often referred to as true stress in the spatial frame. As previously highlighted, in finite deformation analysis the current material configuration at time t + ∆t is unknown and Cauchy stress T at time t + ∆t can not be determined by adding increments at each time-step. This is because Cauchy stress at time t + ∆t is undergoing different configurations as result of rigid body motion, such as translations and rotations [12]. Consequently, in finite deformation with dominated material response, an alternative stress measure must be employed that can track the stress variation from the initial configuration where properties of the material body are known [19]. Utilizing reduced Clausius-Duhem inequality, second Piola-Kirchhoff stress measure can be derived. Stress measure is a function of elongation of material, strain. Therefore Clausius-Duhem inequality as a function of Green-Lagrange strain is introduced as ! ∂ψ ψ˙ = tr E˙ G−L (42) ∂EG−L where Helmholtz free energy is expressed as a function of Green-Lagrange strain as ψ(EG−L) to describe energy variation in the system. Recalling (15) where rate of variation in Green-Lagrange strain is expressed in terms of symmetric part of the velocity gradient and together with the equation above inserted into the reduced Clausius-Duhem inequality in (41). The following is obtained. ! ∂ψ tr ρ E˙ G−L + tr TF −T E˙ G−L F −1 ≥ 0 (43) ∂EG−L Variation in Green-Lagrange strain is arbitrary when the material body undergoes motion, and as previously stated, in accordance to internal energy balance, the temperature is assumed to be constant, thus θ˙ = 0. In this respect, entropy, which measures level of disorder in a thermodynamic system, for instance temperature variation, is neglected.     ∂ψ    −1 −T  ˙ G−L tr −ρ G−L + F TF  E  ≥ 0 (44)  ∂E   | {z } =0 In order to fulfill the inequality above, the under-braced term must become zero, considering rate of change in Green-Lagrange strain E˙ G−L cannot be zero due to large displacements, rotations and translations assumption. Hence, the inequality above can be satisfied by the following equilibrium.

∂ψ ρ = F −1 TF −T (45) ∂EG−L where Cauchy-stress measure T is associated with the current configuration at time t+∆t, at which material properties are unknown. Green-Lagrange strain is associated with the initial configuration where properties of the material body is known, Figure5. The equation above represents rate of change in internal energy with respect to Green-Lagrange Chapter 2. Finite strain theory 30 strain EG−L equivalent to a transformed stress measure defined in the reference frame, i.e. Cauchy stress T and corresponds to ∂ψ T = F ρ F T (46) ∂EG−L The transformed Cauchy stress, i.e. F −1 TF −T corresponds to a stress measure defined in the reference frame. This stress measure is referred to as second Piola-Kirchhoff stress

SX and defined as ∂ψ −1 −T ρX = detF F TF = SX (47) ∂EG−L Recalling the expression for mapping the density from initial to current configuration from Mass balance in section (2.3.1.1), that is (29), allows expressing Second Piola-Kirchhoff Stress in terms of the deformation gradient F . ∂ψ SX = ρX G−L ∂E (48) −1 −T SX = detF F TF

The equation above presents an expression for stress measure in the current configuration, from initial configuration. However, Cauchy stress is unknown, since properties of the material body at time t + ∆t is unknown. Therefore, Cauchy stress T is expressed in terms of second Piola-Kirchhoff stress SX in the aim of satisfying the second axiom of thermodynamics. 1 T T = FSX F (49) detF Second Piola-Kirchhoff stress is compatible with Green-Lagrange strain in a constitutive relation, and its linearization yield

h G−L i ∂SX G−L G−L ∆ SX (E ) = : ∆wE = DfX : ∆wE ∂EG−L (50) ∂SX DfX = ∂EG−L

th where DfX is a 4 order constitutive material stiﬀness tensor representing material behaviour, which is dependent on Lamé parameters µ and λ in case of isotropic conditions.

For an isotropic material such as steel, DfX depends on Young’s modulus E and eﬀect of deformation (expansion and contraction), known as Poisson’s ratio v. For an orthotropic material such as wood, the stiﬀness tensor depends on material properties Young’s modulus E, Poisson’s ratio v and shear modulus G in three directions.

Corollary 2.3.1 Double dot product two tensors is contraction of the tensors with respect to the last two indices of the ﬁrst tensor and the ﬁrst two indices of the second tensor.

a : b = aijbij Chapter 2. Finite strain theory 31

The deﬁnition of Second-Piola Kirchhoﬀ stress, similarly to Hooke’s law can be expressed as G−L SX = DfX : E (51)

th where : denotes contraction due to the fact that DfX is a 4 order tensor. In continuum mechanics of ﬁnite deformation, a set of stress measures are available such as corotated stress tensor, Mantel stress tensor, corotated Cauchy Stress Tensor. However, Second-

Piola Kirchhoﬀ stress tensor SX inTL formulation and Cauchy stress tensor T inUL formulation, will be utilized in this study.

2.4 Principle of virtual work

Finite deformation mechanics depends on principle of internal and external work. This principle considers an imaginary small displacement δw in material particles within the body itself. The virtual displacement results in variation in the body from reference to spatial frame. Virtual displacement δw may not occur physically, but it is needed to derive the principle of virtual work. As the material particles move through virtual displacements, external forces do internal virtual work δW . The external applied forces may consist of surface traction and body forces. According to the Cauchy stress theorem, traction over a surface is equivalent to Cauchy stress multiplied by the out-wards drawn normal vector n. Stress results in internal material response f int and Cauchy stress T is deﬁned in the spatial frame, recognized as true stress (force per unit of area). The material response can be formulated in terms of Cauchy stress T as Z Z t dS = T n ds (52) s s where the surface traction t is integrated over the reference surface is equivalent to a product of Cauchy stress T and out-drawn normal n to the surface, integrated over the spatial surface. Considering principle of virtual displacement in the material particles yield Z Z δw t ds = T δw n ds (53) s s The internal response over the reference surface S, on which traction t is applied, can be transformed to an internal response over a volume, Figure6. Z Z Z δw t ds = T δw n ds = div(T δw) dv (54) s s v | {z } where dv is volume in the spatial frame, likewise for Cauchy stress T . Re-arrangement of the underbraced term in the equation above yields

div(T δw) = tr [T (grad(δw)] + δwdiv(T ) (55) Chapter 2. Finite strain theory 32

Inserting the expression above into (54) results in the following, noting that the term div(T δw) and the volume integral cancel out. Z Z Z δw t ds = tr [T (grad(δw)] dv + δwdiv(T ) dv (56) s v v Recalling linear momentum balance in (30), and considering variation in acceleration of the displacement with respect to time, results in

ρw¨ = div(T ) + ρb (57)

Considering virtual displacement δw allows assumption of zero acceleration of the displacement (w¨ = 0). δwdiv(T ) = −δwρb (58)

Inserting the expression above into Equation (56) results in Z Z Z tr [T (grad(δw)] dv = δwρb dv + δw t dv v v s | {z } | {z } δWint δWext (59)

δWint = δWext

The fundamental principle of virtual work is based on the fact that all internal work are equivalent to external work. As previously highlighted, external work comprises applied loads and surface traction, whereas internal work comprises material response in form of stress (force per unit of area), Figure6.

2.4.1 Internal virtual work

Large deformation mechanics comprise infinitesimal variation of displacements as a material body undergoes motion. It has been highlighted that all internal work done by a continuum system is equivalent to external work. Based on this analogy, it is possible to formulate mechanics of finite deformation in terms of internal work. The internal virtual work in (59) can be expressed in terms of Cauchy stress T and a compatible strain measure EA in (25). Z Z h Ai δWint = tr [T (grad(δw))] dv = tr T δE dv (60) v v In the equation above, the trace operator is unaffected by any eventual asymmetry of (grad δ(w)). The virtual internal work can be further rewritten in terms of Second-Piola G−L Kirchhoff stress SX and Green-Lagrange strain E , in the aim of working with known quantities. Z h Ai δWint = tr det(F )T δE dV0 (61) V0 Mapping of material volume from reference to spatial frame may be derived according to Equation (29) of mass balance. The equation comprises density relationship between Chapter 2. Finite strain theory 33 initial and current configuration. Based on the same analogy, an expression for volume transformation can be derived as

dv = det(F )dV0 (62) where dv is volume in the current configuration and dV0 is volume in the initial configuration and the deformation gradient F includes displacement gradient of element nodes. The transformation has been done in the aim of tracking material properties through deformation from the reference frame where initial material properties are known. Entropy inequality in Section (2.3.1.4), yielded a relationship between Cauchy stress T associated with current configuration, and Second Piola-Kirchhoff stress SX associated with initial configuration, Equation (49). Also in (26), a relationship between Almansi strain EA associated with current configuration, and Green-Lagrange strain EG−L associated with initial configuration was established. These relationships can be inserted into Equation (61), in the aim of expressing virtual variation of internal work in the initial configuration, where properties of the material body is known.   Z    1 T −T G−L −1 δWint = tr det(F ) FSX F F δE F  dV0   V0  det(F ) | {z }  | {z } I (63) I Z h G−L −1i = tr FSX δE F dV0 V0

Corollary 2.4.1 Inside a trace operation, the inner components can be re-arranged. tr(AB) = tr(BA)

Equation (63) can be re-written considering the corollary above as   Z −1 G−L δWint = tr FF SX δE  dV0 (64) V0 | {z } I

Z Z h G−Li h G−L i δWint = tr SX δE dV0 = tr δE SX dV0 (65) V0 V0

It is important to note that Second Piola-Kirchhoﬀ stress measure SX in the above equation, is a function of virtual variation in Green-Lagrange strain δEG−L. Chapter 2. Finite strain theory 34

2.5 Linearization

Solving non-linear partial differential equations requires a consistent solving scheme. Analytical closed-form mathematical solutions in this regard are rather complicated and impracticable. Thus, a sufficient numerical approach with high accuracy and considerable computation speed is desired to employ. This technique requires a consistent linearization to replace the non-linear problem with a series of linear problems, which can be solved through a number of iterations. The systematic linearization procedure is based on the concept of directional derivatives, for which mathematical derivations are out- Figure 7 – Linearization. lined in [5]. The linearization procedure is often referred to as linear approximation, Figure7. Linearization of an arbitrary non-linear function F = F(w) expressed in material coordinate in an incremental approach may be constructed in terms of Taylor’s expansion. F(w, ∆w) = F(w) + ∆F(w, ∆w) + O(∆w) (66) where high order terms are neglected and the linearization operator ∆(•) is a linear function, similar to the virtual operator δ(•), for which usual differentiation properties are valid. The quantity ∆(•) in an approximate numerical solution represents increment of the displacement field w at the each state of equilibrium. The Landau order symbol O(∆w) corresponds to an infinitesimal error, which is considered to approach zero at lim O(∆w)/|∆(w)| = 0. ∆w→∞ In Newton’s solution method, the first term of in the equation above represents a constant solution of the displacements w at the current state. The second term is a linear variation in the displacement function F due to the incremental displacement at the current state ∆w. The directional derivative of F at any state in direction of the displacement increment ∆w can be formulated as

∆F(w, ∆w) = D∆wF(w) (67) where the linear Gateaûx operator D(•) is operating on the increment displacements ∆w and ∆F(w, ∆w) is the linear approximation of the function F with respect to the displacement ﬁeld w,[6]. Chapter 2. Finite strain theory 35

2.5.1 Linearization of virtual strains

In the aim of determining an approximate solution for virtual strains at any arbitrary configuration, a linearization operator is employed. The deformation gradient’s linearization yield ∆F = GRAD(∆w) (68) likewise ∆F −1 = F −1 grad(∆w) (69) where operator (GRAD) is related to initial and (grad) to current configuration. Lin- earization of virtual variation in Green-Lagrange strain in (20), associated with initial configuration follows 1 h iT 1 ∆EG−L = F T GRAD(∆w) + F T GRAD(∆w) 2 2 (70) h i = sym F T GRAD(∆w)

Linearization of Almansi strain in (25), associated with current conﬁguration, yield 1 1 ∆EA = [grad(∆w)]T + grad (∆w) 2 2 (71) = sym [grad (∆w)]

In large deformation analysis linearization of the ﬁrst virtual variation of linearized strain is also required such that h i h i ∆ δEG−L = ∆ sym F T GRAD(δw) h i (72) = sym (GRAD(∆w))T GRAD(∆w) where ∆F T = [GRAD(δw)]T , and ∆ [GRAD (δw)] = 0, since virtual displacement δw is independent of ∆w.

2.5.2 Linearization of virtual work

Virtual variation in internal energy with respect to reference frame i.e. Lagrangian (material) coordinates X in an incremental solution is given in terms of Green-Lagrange G−L strain E and second Piola-Kirchhoﬀ stress SX . In the same manner as expressed in Equation (65), where the expression is integrated over the reference volume. From (65) Z h G−L i δWint = tr δE SX dV0 (73) V0

Linearization of virtual work ∆(δWint) according to [6], yield Z h G−Li ∆ [Wint] = tr SX ∆ δE dV0 V0 Z (74) h G−L i + tr δE ∆(SX ) V0 Chapter 2. Finite strain theory 36

2.6 Static formulations

2.6.1 Total Lagrangian Formulation

Total Lagrangian formulation is expressed in terms of Lagrangian measures of stress and strain, in which derivatives and integrals are taken with respect to Lagrangian (material) coordinates X. In Lagrangian approach, mesh nodes are coincident with material (Lagrangian) points. Hence, Lagrangian coordinates X are constant in time, [2].

Initial conﬁguration X Current conﬁguration x

Figure 8 – Illustration of Lagrangian mesh.

Variation of stress in a material body under motion is associated with variation in strain. Likewise for engineering stress in classical Hooke’s law. InTL formulation, the second G−L Piola-Kirchhoﬀ stress SX is a function of Green-Lagrange strain E . Previously, second G−L Piola-Kirchhoﬀ stress as a function of Green-Lagrange strain, SX (E ), is linearized in Equation (50) as

h G−L i ∂SX G−L G−L ∆ SX (E ) = : ∆E = DfX : ∆E ∂EG−L (75) ∂SX DfX = ∂EG−L

h G−L i The linearized term ∆ SX (E ) in the Equation above, is inserted into Equation (74). The aim is to express virtual variation in internal work in terms of total Second Piola- Kirchhoﬀ stress, as a function of Green-Lagrange strain. Z h G−Li ∆ [δWint] = tr SX ∆ E dV0 V0 Z (76) h G−L G−Li + δE : DfX : ∆E dV0 V0 Inserting previously derived expressions for linearized virtual variation in Green-Lagrange strain ∆(EG−L) in (72) and virtual variation in Green-Lagrange strain δEG−L in (20) and the linearized Green-Lagrange strain ∆EG−L in (70), into linearized virtual variation of internal work ∆[δWint] in (74), yield the ﬁnal mathematical formulation of the Total Lagrangian.

Z h T i ∆ [δWint] = tr GRAD(δw)(GRAD(∆w)) SX dV0 V0 Z (77) h T T i + F GRAD(δw): DfX : F GRAD(∆w) dV0 V0 Chapter 2. Finite strain theory 37

2.6.2 Updated Lagrangian Formulation

Updated Lagrangian formulation is expressed in terms of Eulerian measures of stress and strain, in which derivatives and integrals are taken with respect to spatial (Eulerian) Eulerian coordinates x. Unlike Lagrangian coordinates (mesh), the Eulerian mesh is not constant in time and the boundary nodes do not coincide with material points. Hence, the computation of Updated Lagrangian starts from time t. The diﬀerence between Lagrangian mesh and Eulerian mesh is shown below, Figure9.

Initial conﬁguration X Current conﬁguration x

Figure 9 – Illustration of Lagrangian (L) and Eulerian (E) mesh.

Linearized virtual variation in internal work ∆ [Wint] in Equation (74) considered as Z h G−Li ∆ [Wint] = tr SX ∆ δE dV0 V0 Z (78) h G−L i + tr δE ∆(SX ) dV0 V0 The constitutive relation in (75) between stress and strain in initial conﬁguration is considered as G−L ∆SX = DfX : δE (79) Equation (49) presets relation between Cauchy stress T and second Piola-Kirchhoﬀ stress

SX as 1 T T = FSX F (80) J where J = detF . Linearization of Cauchy stress T in (80) yield

1 T ∆T = F (∆SX )F (81) J Kirchhoﬀ stress T can be expressed in terms of Cauchy stress, also known as true stress T as T = JT (82) Inserting (81) into the above constitutive relation yield the linearized Kirchhoﬀ stress as

T ∆T = F (∆SX )F (83) Chapter 2. Finite strain theory 38

Linearization of displacement, mapping from current to initial conﬁguration, through deformation gradient in terms of Equation (11) is given as

grad(∆w) = GRAD(∆w)F −1 (84)

Linearized Green-Lagrange strain EG−L is given in (70) as h i ∆EG−L = sym F T GRAD(∆w) (85)

The linearized Green-Lagrange strain in (85) and mapping displacement between current and initial conﬁguration in (84), together with the constitutive relation in (79) can be utilized to derive a linearized expression for Kirchhoﬀ stress ∆T as

T ∆T = F (∆SX )F G−L T = F (DfX : ∆E )F T T (86) = F DfX : F GRAD(∆w) F T T = F DfX : F grad(∆w)F F where in last row of the equation above, grad(∆w)F = GRAD(∆w) is obtained by multiplying (84) from both sides by F and inverse rules declared in Corollary (2.2.2) are operating. The linearized Kirchhoﬀ stress ∆T can be further expressed in terms of material properties in the current conﬁguration as

∆T = JDfx : grad(∆w) (87) where constitutive matrix Dfx is obtained through mapping the constitutive matrix in initial conﬁguration DX through jacobian and the deformation gradient F as

−1 Dijkl = J FimFjnDmnpqFkpFlq (88)

th where Dijkl is a 4 tensor and consists of 81 components at decomposition. The linearized

Cauchy-stress in terms of the constitutive matrix Dfx in the current conﬁguration can be expressed as

∆T = Dfx : grad(∆w) (89) As previously ascertained, the mapping between initial and current conﬁguration is performed through the deformation gradient F . In this regard, a push forward of Green- Lagrange strain EG−L can be carried out to determine strain in the current conﬁguration, known as Almansi strain EA, Section (2.2.2)

δEA = F −T (δEG−L)F −1 (90)

The Almansi strain can be linearized in the same manner as Green-Lagrange strain in (72) as h i ∆ δEG−L = F −T (δEG−L)F −1 (91) = F −T GRAD(∆w)T GRAD(δw)F −1 Chapter 2. Finite strain theory 39

Considering mapping of the displacements in (84), gradient of the linearized displacement can be expressed in the same manner as

GRAD(δw) = grad(δw)F −→ GRAD(∆w)T = F T (grad(∆w))T (92) where transpose rule in Corollary (2.6.1) is operating.

Corollary 2.6.1 Product of a transpose matrix and non-transpose matrix is equivalent to switched matrices with swapped transpose. ABT = BAT

The linearized Green-Lagrange strain in (91) can be moreover expressed in terms of the current conﬁguration, considering relation in (92) as h i ∆ δEA = F −T F T (grad(∆w))T grad(δw)FF −1 (93) = (grad(δw))T grad(δw)

A push forward in stresses can be considered in order to obtain stresses in the conﬁgura- tion, known as Cauchy stress T , in terms of Kirchhoﬀ stress T as

T T = JT = FSX F (94)

Volume of the material body in current conﬁguration can be determined through a Jaco- bian matrix, deﬁned as determinant of the deformation gradient, J = det(F ), Equation (317). Hence, the current volume can be achieved as

dv = JdV0 (95) where dv and dV0 are volume of the material body in the current and initial conﬁguration. Virtual variation in internal energy inUL formulation, expressed in terms of Cauchy-stress T and virtual variation in Almansi strain δEA is expressed as Z A δWint = tr T δE dv (96) v Linearization of the virtual variation in internal energy, given above in current conﬁgura- tion, yield Z Z A A ∆ [δWint] = tr T ∆(δE ) dv + tr δE ∆T dv (97) v v Utilizing constitutive relation between linearized Almansi strain, Cauchy stress and displacement in Equations (93), and (89) allow expressing virtual variation in internal energy in terms of displacement as Z Z ∆ [δWint] = grad(δw): grad(∆w)T dv + grad(δw): Dfx : grad(∆w) dv (98) v v Chapter 2. Finite strain theory 40

2.7 Dynamic formulations

In previousTL andUL formulations, the effect of inertia (mass) and damping are not considered. Hence these formulations are recognized as static formulations. In static analysis, the loading is considered as fixed and stationary with a constant magnitude. When loading of a structure varies with time, the inertia load and damping become significant. Consider an offshore, a wind turbine or bridge structure exposed to severe wind load with varying magnitude. This results in oscillation of the structure, where damping plays a key role in decaying the disturbance. Hence, the analysis must recognize impact of mass and damping. Therefore, these formulations are recognized as dynamic formulations. Non-linear dynamic analysis is carried out according to implicit and explicit methods. The implicit method is adequate for problems with prolonged duration effects. Such as dynamic effects of earth quake or large deformation under severe wind loads. Contrast- ingly, the explicit method is adequate for problems with short-term effects, like short term wave propagation. This work, recognizes adequacy of implicit method for analysis of large deformation under dynamic load. As previously highlighted, the dynamic response of materials must consider impact of mass and damping in addition to material stiffness. Material mass contains density of the material while damping is a conversion of mechanical energy under vibration into thermal energy. The higher damping coefficient a material has, the shorter time period of vibration it experiences. Alternatively, damping coefficient is a measure describing how rapidly oscillations decay.

(a) 0.02

0.01

0 1 5 10 (b)

0.008 Displacement [mm]

0.004

0 1 5 10 Time [s]

Figure 10 – (a) Undamped and (b) damped material response under dynamic load. Chapter 2. Finite strain theory 41

The general expression of governing equation of motion for a material undergoing dynamic load is expressed as Ma¨ + Ca˙ + Ka = f(t) (99) where the left hand-side comprises mass matrix M , nodal acceleration a¨, damping matrix C, which is a material property and nodal velocity a˙ . The matrix K is material stiﬀness matrix and a function of nodal displacement a in large deformation analysis. On the left hand-side the external applied load vector f(t) is expressed as a function of time. In large deformation analysis, the equation of motion is expressed in terms of internal forces, which is a product of the tangent stiﬀness matrix KT and displacement vector a, and external forces as f ext.

Ma¨ + Ca˙ + KT a = f ext (100) | {z } f int To solve the equation of motion, initial conditions are declared such that, in the initial state kinematics are known. The material could be at rest or displacements and velocities in the initial state are known. Considering material at rest in the initial state, implies that the material neither experiences displacement nor velocity.

a(0) = a0 a˙ (0) = a˙ 0 (101)

Dynamic response of structures exposed to heavily wind-induced or earthquake vibration, is often analyzed under a particular displacement. On the basis of which, appropriate damping solution is determined to reduce the resonant vibrations. In this manner, the displacement a(n) and corresponding internal forces f int(n) at the desired state shall enter the equation of motion as initial conditions. For better understanding of deﬁnition of velocity and acceleration under motion in large deformation analysis, the reader is advised to review equation (4). Variation of damping ratio in an oscillating material body under implicit time-integration method, can be modelled through Rayleigh viscous damping. This damping model is proportional to a linear combination of material mass and tangent stiﬀness, such that

C = d1M + d2KT (102) where d1 and d2 are constants of proportionality. In non-linear dynamic analysis, stability of the time-integration algorithm must be given a special attention. An algorithm is recognized as conditionally stable, if the stability of the numerical algorithm rests upon how ﬁne the mesh is. Alternatively, the number of employed time-steps shall not exceed critical time-step ∆t < ∆tcr. Contrastingly, a solution algorithm is designated as unconditionally stable if stability of the algorithm does not depend on number of time-steps. In the aim of integrating the equation of motion of a Multi Degree of Freedom (MDOF) Chapter 2. Finite strain theory 42 system, numerical algorithms are combined with time-integration methods. In an implicit approach, a number of time-integration methods are proposed by [4] and [23]. Such as Central Diﬀerence Method, Newmark-β Method and Wilson-θ Method, Energy Conserv- ing Algorithm. In this work, the unconditionally stable Newmark-β and Energy conserving Algorithm time-integration methods will be considered. The mentioned methods are extracted from [23] and [4].

2.7.1 Newmark-β method

In Newmark-β method the velocity in the current state is determined through time- integration over the acceleration as

Z tn+1 a˙ n+1 − a˙ n = a¨(t) dt (103) tn where the subscript n indicates the current state at time t, and the acceleration is determined through diﬀerentiation of velocity with respect to time as ∂a˙ = a¨ (104) ∂t In the Newmark-β method, the updated velocities are determined as

a˙ n+1 − a˙ n = (1 − γ)∆ta¨n + γ∆ta¨n+1 (105) a˙ n+1 = a˙ n + ∆t [(1 − γ)a¨n + γa˙ n+1] where ∆t = tn+1 − tn is size of the integration at each time step, and 0 ≥ γ ≤ 1 is a stability parameter. The Newmark equation can be utilized to establish an expression for the updated displacements as

a˙ n+1 − a˙ n (1 − γ)∆ta¨n γ∆ta¨n+1 = + (106) ∆t ∆t ∆t | {z } an+1−an Newmark has originally proposed an unconditionally stable Constant Average Accelera- 1 tion Method, in which case the introduced stability parameter is given as γ = 2 . Hence the updated displacements can be expressed as ∆t ∆t an+1 − an = a˙ n + a˙ n+1 (107) 2 2 where the updated displacements an+1 are a function of the updated velocities, which are unknown at state n + 1. In the aim of establishing a constitutive relation for the updated displacements in terms of known quantities, the equations (105) and (107) are combined as 2 ∆t ∆t ∆t ∆t  a˙ n+1 = a˙ n + (1 − γ)∆ta¨n + γ a¨n+1  2 2 2 2 (108)  ∆t ∆t a˙ n+1 = an+1 − a˙ n − an 2 2 Chapter 2. Finite strain theory 43 where the updated velocities are set in equilibrium. Hence, the updated displacements become ∆t ∆t ∆t2 an+1 = an + a˙ n + a˙ n + [(1 − γ)a¨n + γa¨n+1] 2 2 2 (109) | {z } ∆ta˙ n As previously declared, the equation of motion is desired to be solved implicitly at an updated step n + 1 as

Ma¨n+1 + Ca˙ n+1 + KT an+1 −f ext(n + 1) = 0 (110) | {z } f int The updated displacements at state n+1 in the Newmark-β method is expressed in terms of an additional parameter β. For an unconditionally stable scheme, Newmark proposed as previously highlighted, Constant Average Acceleration Method, in which case γ = 2β.

Hence the updated displacements an+1 can be expressed as ∆t2 an+1 = an + ∆ta˙ n + [(1 − 2β)a¨n + 2βa¨n+1] (111) 2 which reminds of a Taylor expansion. The equation above contains two unknown quantities, the updated displacements an+1 and accelerations a¨n+1. Hence, an expression for the updated accelerations are derived from the given equation and its components are sorted in terms of Newmark constants as C1,C2,C3 as 1 1 1 1 − 2β a¨n+1 = an+1 − an − a˙ n − a¨n (112) β∆t2 β∆t2 β∆t 2β | {z } | {z } | {z } | {z } C1 C1 C2 C3

The updated accelerations at a step forward can be simply expressed as

? a¨n+1 = C1an+1 − a¨n (113)

? where a¨n correspond to ? a¨n = C1an + C2a˙ n + C3a¨n (114) The expression above can be inserted into Equation (105) yield

a˙ n+1 = a˙ n + ∆t(1 − γ)a¨n + ∆tγC1an+1 − ∆tγa¨n

(((( a˙ n+1 = a˙ n + (∆(t(1(− γ)a¨n + ∆tγC1an+1 − ∆tγC1an − ∆tγC2a˙ n − ∆tγC3a˙ n (115)

a˙ n+1 = ∆tγC1an+1 − ∆tγC1an − (1 − ∆tγC2)a˙ n

1−γ where it is noted that C3 correspond to γ . The values of C1 and C2 are inserted into the ﬁnal form of the updated velocities a˙ n+1 and additional Newmark constants are introduced as ¨∆¨tγ ¨∆¨tγ ¨∆¨tγ a˙ n+1 = an+1 − an − (1 − )a˙ n β¨∆¨t2 β¨∆¨t2 β¨∆¨t γ γ γ (116) a˙ n+1 = an+1 − an − (1 − ) a˙ n β∆t β∆t β | {z } | {z } | {z } C4 C5 C5 Chapter 2. Finite strain theory 44

The updated velocities at a step forward can be simply expressed as

? a˙ n+1 = C4an+1 − a˙ n (117)

? where a˙ n correspond to ? a˙ n = C4an + C5a˙ n (118) In conclusion, the updated velocities and accelerations are determined as

? ? a¨n+1 = C1an+1 − a¨n a¨n = C1an + C2a˙ n + C3a¨n (119) ? ? a˙ n+1 = C4an+1 − a˙ n a˙ n = C4an + C5a˙ n where the constants in summary are given as 1 1 1 − 2β C1 = C2 = C3 = β∆t2 β∆t 2β (120) γ γ − β C4 = C5 = β∆t β

Energy conservation and stability of Newmark-β can be examined by multiplying the equation of motion in (110) by a˙ T .

T T T T a˙ Ma¨ + a˙ Ca˙ + a˙ KT a = a˙ f ext (121)

Taking time-derivative of the terms associated with mass and stiﬀness in the equation above follows   d 1 T 1 T T T  a˙ Ma˙ + a KT a = a˙ f ext − a˙ Ca˙ (122) dt 2 2 | {z } | {z } | {z } The equation above represents rate of variation in kinetic and internal energy with respect to time in relation with energy dissipation, which can be simply expressed as d [Kinetic energy + Internal energy] = a˙ T f − Energy dissipation (123) dt ext For a system without energy dissipation and external forces, no energy production within the material is expected. Thus, the total energy and its rate of variation must be zero. dE ∆E = 0; = 0 (124) dt where E is the total energy production within the system. The total energy between states n + 1 and n yield

T T ∆E = a˙ M∆a˙ + a KT a = 0 (125)

Consequently, the total energy according to Newmark-β method becomes

tn+1 1 T T ∆E = a˙ M∆a˙ + a KT a 2 tn (126) 1 T = −(γ − )∆a Keﬀ∆a 2 Chapter 2. Finite strain theory 45

where the effective stiffness Keff can be expressed as

1 2 −1 Keﬀ = KT + (β − γ)∆t KT M KT (127) 2 The stability conditions of Newmark parameters is illustrated below, where w respresents modal stiﬀness.

1 Stable complex Stable 2

) 0.75 t ∆ w ( 0.5 / + 1

β 0.25 Unstable

0 0.5 γ 1 1.5

Figure 11 – Stability diagram of Newmark-β Method.

In an implicit analysis with algorithmic damping, the following stability coeﬃcients introduce an unconditionally stable scheme. 1 γ = 2 (128) 1 1 β = γ or β = 2 2 In an explicit time-integration with unconditional stability and algorithmic damping to suppress high frequency components, the following coeﬃcients are suggested by Krenk [14]. 1 γ = β = 0 (129) 2 Chapter 2. Finite strain theory 46

2.7.2 Energy conserving algorithm

In energy conserving, the internal energy Πint is derived in terms of strain energy density function as Z Πint = w(E) dV0 (130) V0

Time derivative of the internal energy Π˙ int follows Z Π˙ int = E˙ : S dV0 (131) V0 The equation of motion for an undamped system may be formulated as

Ma¨ + f int = f ext (132)

It follows that T T T a˙ Ma¨ + a˙ f int = a˙ f ext | {z } ˙ Πint (133) T ˙ T a˙ Ma¨ + Πint = a˙ f ext Integrating the equation of motion allows expressing the internal energy as d 1 T T a˙ Ma˙ + Πint = a˙ f (134) dt 2 ext where the total energy of the system is formulated as

1 T E = a˙ Ma˙ + Πint (135) 2 In energy conserving approach, it is important with nonexistent of internal energy when the system is not energized.

tn+1 1 T a˙ Ma˙ + Πint = 0 (136) 2 tn In numerical analysis the diﬀerential equations are expressed as algebraic equations. The equation of motion (132) can be integrated from time tn to tn+1 as

Z tn+1 m M∆a˙ + f int dt = ∆tf ext (137) tn where ∆t denoted length of the time-step. The superscript m denotes mean-value of the external forces and ∆a˙ is the velocity at current state. The mentioned quantities can be deﬁned as

Z tn+1 Ma dt Ma tn+1 ⇐⇒ M a˙ − a˙ M a˙ ¨ = [ ˙ ]tn ( n+1 n) = ∆ (138) tn

Z tn+1 m f ext dt = ∆tf ext (139) tn Chapter 2. Finite strain theory 47

The integral over the internal forces cannot be taken unless variation in strains are known in time. Hence, the following approximation is introduced

Z tn+1 ? f int dt = ∆tf int (140) tn The equation of motion for an undamped system expressed in terms of the introduced approximation becomes ? m M∆a˙ + ∆tf int = ∆tf ext (141) The displacement at the current state may be formulated in terms of Newmark-β approach in (107), which considers constant Constant Average Acceleration Method. ∆t an+1 − an = (a˙ n+1 + a˙ n) (142) 2 The velocity is formulated as a mean velocity as 1 a˙ = (a˙ n+1 + a˙ n) (143) 2 The relationship above allows expressing the displacement at the current state given in (142) as 1 an+1 − an = ∆t (a˙ n+1 + a˙ n) (144) | {z } 2 ∆a | {z } a˙ Hence, the displacement at the current state becomes

∆a = ∆t a˙ (145) and the velocity at the current state can be derived accordingly. 2 a˙ n+1 + a˙ n = ∆a ∆t 2 a˙ n+1 = ∆a − a˙ n ∆t 2 (a˙ n+1 = ∆a − a˙ n) − a˙ n ∆t (146) 2 a˙ n+1 − a˙ n = ∆a − 2a˙ n | {z } ∆t ∆a˙ 2 ∆a˙ = ∆a − 2a˙ n ∆t The corresponding acceleration a¨ can be determined through diﬀerentiating the velocity with respect to time accordingly.

∂∆a˙ a˙ n+1 − a˙ n ∆a¨ = = ∂t ∆t 1 a¨n+1 − a¨n = (a˙ n+1 − a˙ n) (147) ∆t 1 a¨n+1 = (a˙ n+1 − a˙ n) + a¨n ∆t Chapter 2. Finite strain theory 48

In numerical analysis it is important that summation of acting forces are assigned into a variable called residual reﬀ in the aim of convergence study. The Equation (146) can be substituted into the equation of motion in Equation (141), whereby numerical error reﬀ can be derived. 2 ? ? M ∆a − 2a˙ n + ∆tf = ∆tf (148) ∆t int ext

  2  2 ? ?   M (an+1 − an) −2Ma˙ n + ∆tf int = ∆tf ext (149) ∆t ∆t | {z } ∆a

4 4 ? ? reﬀ = 2 M (an+1 − an) − Ma˙ n + 2f int − 2f ext (150) ∆t | {z } ∆t ∆a The energy conserving properties can be introduced to the system in the aim of ensuring no energy dissipation. The equation of motion given in Equation (141) can be pre-multiplied by transpose of mean value of the velocity given in Equation (143) as

T T ? T m a˙ M∆a˙ + a˙ ∆t f int = a˙ ∆t f ext (151) | {z } | {z } ∆aT ∆aT

T T ? T m a˙ M∆a˙ +∆a f int = ∆a f ext (152) | {z } The underbraced term in the equation above can be further developed considering the following relations

T 1 T T a˙ M∆a˙ = (a˙ − a˙ )M(a˙ n+1 − a˙ n) 2 n+1 n 1 h T T T T i = a˙ Ma˙ n+1 − a˙ Ma˙ n + a˙ Ma˙ n+1 − a˙ Ma˙ n 2 n+1 n+1 n n 1 h T T i = a˙ Ma˙ n+1 − a˙ Ma˙ n 2 n+1 n   (153) 1    ˙ T ˙ T  = (an+1 − an ) M (a˙ n+1 − a˙ n) 2 | {z } | {z } R a˙ T dt R a˙ dt

1 h itn+1 = a˙ T Ma˙ 2 tn The above relation substituted into the equation of motion, Equation (152) yield

tn+1 1 T T ? T m a˙ Ma˙ + ∆a f int = ∆a f ext (154) 2 tn In order to fulﬁll energy conserving properties, all internal forces in a system must be zero in absence of external excitation. The relation in Equation (136) reveals that the following relation holds

tn+1 tn+1 1 T 1 T T ? a˙ Ma˙ + Πint = a˙ Ma˙ + ∆a f int (155) 2 tn 2 tn Chapter 2. Finite strain theory 49

tn+1 aT f ? [Πint]tn = ∆ int (156) The internal forces comprise stress and strain variation in the system under excitation. The internal forces can be formulated in terms of strain energy density function given in Equation (130) as Z T ? ? T ? ∆a f int = (E ) S dV0 (157) V0 How the strains should be formulated is unclear. Hence, St. Venant Kirchhoﬀ material model is considered, for which strain energy is given as 1 w = ET DEf (158) 2 The internal forces for a St. Venant Kirchhoﬀ material becomes

t 1 Z 1 Z n+1 ET DEf dV − ET DEf dV [Πint]tn = n+1 n+1 0 n n 0 2 V0 2 V0 Z (159) T = ∆E S dV0 V0 In formulation of internal forces, the variation in strains ∆Ed are formulated as strains in the current state. However, the stresses are computed based on mean-value of strains between previous and current conﬁguration. 1 ∆E = En+1 − En; S = DfE; E = (En+1 + En) (160) 2 The relations above reveals that the stresses and strains associated with internal forces for St. Venant-Kirchhoﬀ materials represent

E? = ∆E; S? = DfE (161)

Stability in Energy Conserving may be evaluated through expressing equation of motion as state equations   a x˙ = f(x, t) −→ x˙ =   (162) a˙ where f(x, t) represents total energy in the system   a˙ f(x, t) =  −1  (163) −M (f int + Ca˙ − f ext(t))

In the aim of investigating motion perturbation, the motion is subdivided into two motions where xF (t) is the true motion and x(t) is an approximated motion, for which equations of motion are expressed as

F F F Ma¨ + Ca˙ + f int(a ) − f ext(t) = 0 (164) Ma¨ + Ca˙ + f int(a) − f ext(t) = 0 Chapter 2. Finite strain theory 50

Hence, the disturbance of the system can be formulated as a combination of true internal response and a magnitude of velocity and acceleration that perturbs the system response motion as P P F Ma¨ + Ca˙ + f int(a) − f int(a ) = 0 (165) A truncated Taylor’s expansion may be employed where high-order terms are neglected, to establish an equilibrium between the true and perturbed motion

F P f int(a) − f int(a ) = KT a (166) where the tangent stiﬀness matrix is computed for the true motion of the system. Based on the introduced analogy the linearized equation of motion for the perturbation of the motion can be expressed as

P P P Ma¨ + Ca˙ + KT (a ) = 0 (167)

The corresponding relation in state space becomes     P a 0 I x˙ =   A =  −1 −1  (168) a˙ −M KT M C

Stability of the system can be archived through taking eigenvalue of the system. However, the solution can also be simply expressed as solution of PDE with ansatz b as

aP = be(λt) (169)

The corresponding solution becomes

2 λ M + λC + KT b = 0 (170)

The eigenvalue problem above may result in a complex number. Thus, a conjugated eigenvector b is deﬁned, by which the system is pre-multiplied. Hence, the eigenvalue problem simpliﬁes to the following.

2 T T T λ b Mb + λb Cb + λb KT b = 0 (171)

Solving the quadratic equation above using quadratic formula, the eigenvalue λ becomes v u 2 T u T T b Cb u b Cb b KT b λ = − T ± t T  − T (172) 2b Mb 2b Mb b Mb

The equation above implies that the motion is stable if the real part of the eigenvalue λ T is less than zero, i.e. Re(λ) < 0. This is the case when the tangent stiffness term b KT b becomes greater than zero. Oppositely, when the stiffness becomes negative the motion becomes unstable. Alternatively, only if the smallest eigenvalue of the stiffness term λK is less than zero λK < 0 the motion is stable. It is important to bear in mind that an unstable system can not be stabilized through influence of damping. Chapter 2. Finite strain theory 51

2.8 Constitutive material behaviour

In ﬁnite strain theory, elastic materials are subdivided into two main categories, elastic and hyperelastic. These material models are recognized for their elastic kinematics behaviour under deformation. The elastic materials may be linear or non-linear materials with isotropic or anisotropic behaviour. Hyperelastic materials are in-compressible materials with ability to change shape while keeping the T overall volume the same. This material model char- acterizes by highly non-linear stress-strain relation,

Elastic stiff and soft material behaviour under compres- Hyperelastic sion and tension. The hyperelastic materials are commonly utilized when high flexibility is required E such as bridges and fluid seals. Hyperelasticity theory concerns deriving stress-strain relationship from Figure 12 – Stress-strain curve. strain energy density function. The choice of constitutive model for a hyperelastic material may be strongly associated with mechanical behaviour of the material. In modeling of elastic materials a wide range of constitutive models are available. Examples of which, are

Neo-Hookean (Reduced ﬁrst order polynomial)

Mooney-Rivlin (Complete ﬁrst order polynomial)

Saint Venant–Kirchhoﬀ (First order polynomial)

The Neo-Hookean and Mooney-Rivling material models are appropriate for hyperelastic materials, for example rubber-like materials or polymers. The Saint Venant-Kirchhoﬀ material model is commonly utilized in analysis of elastic materials. New material models may be constructed based on mechanical behaviour of a certain material based on numerical simulations or experiments. This work considers Saint Venant–Kirchhoﬀ model, which is considered as an extension of geometrically linear elastic material model to geometrically nonlinear regime.

Materials are classified and categorized according to their behaviour in different directions. Isotropic materials are characterised by identical mechanical and thermal properties in all directions. Hence, defining such a material only requires 2 independent elastic constants, often Young’s modulus E and Passions ratio ν in one direction. Example of isotropic materials are metals and glasses. These materials may be studied in 2D plane strain or plain stress. Chapter 2. Finite strain theory 52

Anisotropic materials are distinguished by unique and independent mechanical and thermal properties in all directions. Defining such a material requires 21 independent elastic constants in different directions. Examples of these type of materials are wood, compos- ites and biological tissues. Orthotropic materials are a subset of anisotropic materials with unique and independent material properties in three perpendicular directions. Hence, defining these materials requires 9 independent elastic constants.

Anisotropy Orthotropy

Figure 13 – Illustration of anisotropy and orthotropy.

Wood has strongly shown orthotropic behaviour in axial, radial and circumferential r directions due to orientation of its fibers. Hence, in defining wood as an orthotropic material in a Cartesian coordinate system, three perpendicular axis are introduced. Longitudinal (axial) direction is parallel to Fiber direction the grain, radial and tangential axis are t perpendicular to the grain direction. These axis makes it possible to assign appropriate material constants associated with each direction. It is important to emphasize that l wood is as strongest in its fiber direction Figure 14 – Orthotropic directions. and as weakest perpendicular to the fiber direction, which must be defined in the material constitutive stiffness matrix DfX . Due to orthotropy behaviour of wood it may only be appropriate to study wood based materials in 3D, which has been taken into account in this work. Chapter 2. Finite strain theory 53

2.8.1 Orthotropic coordinate transformation

Orthotropic materials with unique and independent properties in different directions require coordinate transformation in the aim of assigning appropriate properties. In a FEM approach a domain is often meshed with a great number of unstructured and orientated infinitesimal elements. For each of these elements appropriate material properties must be assigned in all directions. In the case of wood, appropriate material properties in radial, tangential and longi- x2 tudinal directions must be assigned to each ? D? x2 ijkl element. For example wood’s pith, around which annual rings are located, may be lo- x? cated anywhere inside the cross section or 1 may be outside. Hence, local coordinates of each element with respect to annual ring x1 direction at the desired element must be Dijkl transformed to a stationary global coordinate system. It is important to note that the local coordinate system must be con- x structed fulfilling the criteria that the tan- 3 gent axis is actually tangent to the annual Figure 15 – Coordinate transformation of ring. In the aim of defining appropriate annual rings in wood. material properties, the constitutive material property matrix defined in the local coordinate system, denoted by "star", has to be transformed to the "non-star" global coordinate system.

The coordinate transformation of any arbitrary vector between two coordinate systems can be conducted through a rotation matrix Q and a transformation vector c which takes into account distance between origins of the coordinate systems.

? x2 x2 b? x ? x2 x 2 r ? xb xa Q ? a x1 x ? x 1 c α x1 x1

Figure 16 – Illustration of mathematical coordinate transformation. Chapter 2. Finite strain theory 54 where the rotation matrix for an in-plane rotation in Figure 16, is deﬁned as   cos(α) −sin(α) 0     Q = sin(α) cos(α) 0 (173)   0 0 1

Any point deﬁned in "star" marked coordinate system can be transformed to the "non-star" coordinate system as x? = c + Q x (174) where the rotation matrix Q has the following properties

QQT = I (175) Q−1 = QT

The vector r in Figure (16) can be computed through two vectors deﬁned in respective coordinate system, pointing at each end of the vector r as

r = xa − xb ? ? (176) r? = xa − xb and transformation of the vectors between respective coordinate systems are performed as ? xa = c + Q xa ? (177) xb = c + Q xb Hence, the vector r as it is a ﬁrst order vector can be transformed as

? ? r? = xa − xb ⇐⇒ Qxa − Qxb = Q(xa − xb) (178) | {z } r r? = Qr where C matrix cancels out due to deﬁning length of vector r through vectors xa and xb. The transformation above characterises objective law, which is deﬁned by rigid transformation. In this respect, the vector r must not undergo any motion. Transforming second order tensors between coordinate systems may be performed as

S = Ur S? = QS (179) S? = U ?r? r? = Qr where S is a vector and U may be a rotational second order tensor multiplied by vector r. Hence, transformation of S becomes

QS = U ?Qr ⇐⇒ QT QS = QT U ?Qr | {z } I (180) T ? T ? S = Q U Qr ⇐⇒ U r = Q U Qr |{z} Ur Chapter 2. Finite strain theory 55

U = QT U ?Q where the second order tensor U is transformed from "non-star" coordinate system to "star" coordinate system through the rotation matrix Q. In the same manner, Cauchy stress can be transformed between local and global coordinate systems.

T = QT T ?Q (181)

In coordinate transformation a special attention is given to transformation of deformation gradient F and strains E. In Equation (2) it is declared that transformation of a vector between initial and current conﬁguration deﬁned in any arbitrary coordinate system as

dx = F dX (182) dx? = F ?dX where it is noted that the configuration transformation is defined in the same coordinate system. If transformation of the deformation gradient between different coordinate systems is desired, the following may be considered

dx? = Qdx ⇐⇒ dx? = QF dX ⇐⇒ dx? = F ?dX | {z } F dX (183) F ?¨dX¨ = QF ¨dX¨ F ? = QF where it is noted that the deformation gradient F transforms in the the same manner as a vector, Equation (178). Strains tend to have same magnitude independent of coordinate systems. As an example, Green-Lagrange strain EG−L can be transformed as

1 G−L 1 T ? EG−L = F T F − I E? = F ? F ? − I 2 2   1  T T  (184) = F Q Q F − I 2 | {z } I = EG−L where it is noted that I? = I. In coordinate transformation, scalars have the same magnitude independent of coordinate system. For example, temperature in "non-star" coordinate system has the same magnitude as in the local "star" coordinate system.

θ = θ? (185)

The independency of strains of coordinate transformation confirms that transformation of the constitutive material stiffness matrix DfX is sufficient for defining material properties in the global coordinate system with respect to local coordinate system. The transformation may be carried out only in 2D where the pitch may be assumed to continue Chapter 2. Finite strain theory 56 undeviatingly through longitudinal direction of the material. In this manner the rotation matrix may be constructed as     1 0 0 e11 e12 e13     ?     e = 0 cos(α) −sin(α) ⇐⇒ e21 e22 e23 (186)     0 sin(α) cos(α) e31 e32 e33 where the longitudinal axis is fixed. Transformation of the constitutive matrix becomes then ? Dijkl = QmiQnjDmnpqQpkQql (187) The compliance matrix is defined as   Ce1111 Ce1122 Ce1133 Ce1112 Ce1123 Ce1113     Ce2211 Ce2222 Ce2233 Ce2212 Ce2223 Ce2213     Ce3311 Ce3322 Ce3333 Ce3312 Ce3323 Ce3313 Cfx =   (188)   Ce1211 Ce1222 Ce1233 Ce1212 Ce1223 Ce1213     Ce2311 Ce2322 Ce2333 Ce2312 Ce2323 Ce2313   Ce1311 Ce1322 Ce1333 Ce1312 Ce1323 Ce1313 and the constitutive material matrix DfX is defined as

−1 DfX = CfX (189) due to the following relations E = CfX T −1 (190) T = CfX E The coordinate transformation can also be carried out by using a transformation G matrix to avoid tensor decomposition.

x2 e ? 2

x1 e ? 3

e ? x3 1 D D? Figure 17 – Element coordinate transformation Chapter 2. Finite strain theory 57

The transformation matrix is constructed using components of the rotation matrix e as

 2 2 2  e11 e12 e13 e11e12 e12e13 e13e11    2 2 2   e21 e22 e23 e21e22 e22e23 e23e21     2 2 2   e31 e32 e33 e31e32 e32e33 e33e31  G =   (191)   2e11e21 2e12e22 2e13e23 e11e22 + e12e21 e12e23 + e13e22 e13e21 + e11e23     2e21e31 2e22e32 2e23e33 e21e32 + e22e31 e22e33 + e33e32 e23e31 + e21e33   2e31e11 2e32e12 2e33e13 e31e12 + e32e11 e32e13 + e33e12 e33e11 + e31e13 where e is component of the rotation matrix e?. The coordinate transformation in this fashion becomes then ? T DfX = G DfX G (192) ? where DfX is element constitutive matrix deﬁned in the initial conﬁguration, Equation (189). In order to compute the angle θ to construct the matrix e?, whose components enter the transformation G, trigonometrical rules may be considered, Figure 18.

dy θ x2 dx t0

Figure 18 – Coordinate transformation of annual rings in wood in 2D. Chapter 2. Finite strain theory 58

Wood grows naturally and its pith may grow straight or not straight. In wood boards, the pith is often located outside of the cross section and varies along the length of the board depending on sawing considerations, Figure 19. In this regard, the coordinate transformation must both consider rotation along the annual rings and variation in the pith along the board.

Pith

Sawn board

Annual rings

Figure 19 – Illustration of annual rings pith variation along length of a wood board.

In order to determine direction cosines of coordinate transformation, Equation (186), a routine proposed by [24] may be considered. In the mentioned routine the deviation of the pith along the length of the wood, spiral grain as well as conical shape are being taken into account. A stationary coordinate system is established, from which vectors are employed in the aim of determining corresponding element coordinates. The element coordinate is denoted by p, at which the directional cosines with respect to tangential, radial, longitudinal axis are desired. Vectors A and B are employed from the global coordinate system to each end of the pith in the aim of establishing a relationship between the global coordinate system i, j, k and the element coordinate system t0, r0, l0,. The vectors A, B and p from the origin to the point where they are deﬁned in Figure 20 follow the equation of vectors

A = Axi + Ayj + Azk (193)

B = Bxi + Byj + Bzk (194)

p = pxi + pyj + pzk (195) Chapter 2. Finite strain theory 59

B(bx, by, bz) x

S B P (px, py, pz) i l t0 R 0 P k r0 z j Q A y N

A(ax, ay, az)

Figure 20 – Geometry and mathematical description of coordinate transformation of wood in 3D.

The vector N from point A to p, and vector S from the base (point A) to the top (point B) can be computed accordingly.

N = P − A (196)

S = B − A (197) The unit vector q following direction of the pith is deﬁned as S q = (198) |S| Projection of vector N in the direction of the pith yield vector Q.

Q = (N · q)q (199)

Vector R perpendicular to the pith and surface of the cylinder can be computed as

R = N − Q (200)

The element unit vectors t0, r0, l0 can be determined accordingly.

l0 = q (201) Chapter 2. Finite strain theory 60

R r0 = (202) |R|

t0 = l0 × r0 (203)

The given directions coincide with orthographic directions of wood. However, eﬀects such as conical and spiral grain eﬀects are not considered in this work, further can be found in [24]. The given directions can be utilized in the aim of transforming global coordinate system to material local coordinate system accordingly.     l0 i   T     ?   r0 = e j (204)     t0 k where the rotation matrix e? is given as

 x x x    el er et l0x r0x t0x  0 0 0    e? ey ey ey  l r t  =  l0 r0 t0  =  0y 0y 0y  (205)  z z z    el0 er0 et0 l0z r0z t0z

The transformation procedure can be carried out in the same manner as in Equation (192), where components of the transformation matrix are extracted from the above given rotation matrix e?. Chapter 2. Finite strain theory 61

2.8.2 Invariants of tensors

The property of strains after mathematical transformation reveals invariant property. Invariant is a condition that holds for a mathematical object or system to remain unchanged after transformation operations. This property can be utilized in deriving stress- strain relation, i.e. the constitutive material tensor Dijkl. The invariants of a second order tensor is given as

? I1 = tr(T ) = tr(T )

1 2 1 2 1 ? 2 1 ?2 I2 = tr(T ) − tr(T ) = tr(T ) − tr(T ) (206) 2 2 2 2 ? I3 = det(T ) = det(T ) where the invariants are independent of coordinate systems they are deﬁned in. It is important to note that the above given invariants are associated with orthotropic materials. In case of anisotropic materials, additional invariants become operative.

For any Cauchy stress tensor, there is a state of orientation at which Tij = 0 for i =6 j, and T =6 0 for i = j. This implies that at a state of orientation, all off-diagonal stresses are zero, while diagonal terms in Tij are non-zero. The directions corresponding to this stress state are referred to as principal stress state. The principal stress direction corresponding to a unit vector nî and principal stress plane corresponding the normal vector to the plane maybe considered to project Cauchy stress onto the principal normal stress plane. This results in a normal stress corresponding to an eigenvalue λ multiplied by the unit vector nî corresponding to normal stresses n.       T11 T12 T13 n1 n1             T21 T22 T23 n2 = λ n2 (207)       T31 T32 T33 n3 n3

This relationship in a generalized manner may be expressed as

Λnî = λinî = 0 (208) where Λ is a stress matrix desired to be projected onto the principal normal stress plane. The above equations has only one solution at which the following condition must be fulfilled

det(Λ − λiI) (209)

The given condition is often referred to as the eigenvalue problem. Chapter 2. Finite strain theory 62

2.8.3 Constitutive model behaviour

The constitutive behaviour of elastic materials is derived in terms of strain energy density function. In the aim of tracking energy variation within a material body through finite deformation, Helmholtz free energy ψ may be employed. Previously in Equations (42) and (48), Helmholtz free energy as a function of Green-Lagrange strain ψ(EG−L) was employed to characterise energy variation within a material body undergoing finite deformation. In terms of which, an expression for second Piola-Kirchhoff stress SX was derived under the condition of satisfying entropy inequality of thermodynamics. In constitutive modeling of elastic materials behaviour, Helmholtz free energy as a function of invariants may be employed, such that ψ(I1,I2,I3). Inserting invariants in Equation (206) into the Helmholtz free energy function, yields

ψ = ψ(I1,I2,I3) 1 (210) ψ = ψ tr(E), (tr(E))2 − tr(EE), det(E) 2 where E is strain. The invariants can be diﬀerentiated with respect to strain in the aim of expressing strain-energy variation with the material body as

∂ψ(I1,I2,I3) ∂ψ ∂I1 ∂ψ ∂I2 ∂ψ ∂I3 = + + (211) ∂E ∂I1 ∂E ∂I2 ∂E ∂I3 ∂E Hence, the partial derivative of the invariants with respect to strain become

∂I1 ∂(tr(E)) = = I ∂E ∂E h i ∂ 1 tr(E)2 − (tr(E2) ∂I2 2 (212) = = tr(E)I − E = I1I − E ∂E ∂E

∂I3 ∂det(E) = = [det(E)] E−1 ∂E ∂E and the general formulation of strain energy density function becomes

∂ψ ∂ψ ∂ψ ∂ψ ∂ψ −1 = I + I1I − E + I3E (213) ∂E ∂I1 ∂I2 ∂I2 ∂I3 | {z } | {z } | {z } where the under-braced terms indicate components of the strain-energy variation in Equa- tion (211). Stress variation within a material body through finite deformation may be expressed in terms of strain energy density function as ∂ψ SX = ρX ∂EG−L " # (214) ∂ψ ∂ψ ∂ψ ∂ψ −1 = ρX I + I1I − E + I3E ∂I1 ∂I2 ∂I2 ∂I3 As previously declared, this work intends to implement Saint Venant–Kirchhoff constitutive model ψ = ψ(I1,I2) in which the strain energy density function ! λ 2 ψ = + µ I − 2µI2 (215) 2 1 Chapter 2. Finite strain theory 63 where λ and µ are Lamé coefficients. Differentiating the strain energy density function with respect to invariants yield ∂ψ = λI1 + 2µI1 ∂I1 ∂ψ = −2µ (216) ∂I2 ∂ψ = 0 ∂I3 Consequently, second Piola-Kirchhoff stress in Equation (211) is expressed in terms of Saint Venant–Kirchhoff constitutive model for strain energy density function, Equation (213), as ¨¨ ¨¨ SX = ρX [(λI1 + Ï12µ − Ï12µ)I + 2µE] (217) SX = λI1I + 2µE In the equation above, the material is considered as solid and homogeneous thus the material density ρX is set to one. The expression above in terms of index notations may be expressed as

SXij = λ δijEkk +2Eij (218) | {z } Itr(E) where the new quantity δij is referred to as Kronecker delta and has the following properties  0 if i =6 j δij = (219) 1 if i = j Kronecker delta is a piecewise function of variables i and j in the Euclidean space. In continuum mechanics the Lamé coefficients λ and µ are material quantities arising in stress-strain relationships and are frequently referred to as Young’s Modulus and Shear Modulus. In the aim of compatibility with general formulation of material constitutive matrix Dijkl the followings are considered νE λ = . E: Elastic Modulus (1 + ν)(1 − 2ν) (220) E µ = . G: Shear Modulus 2(1 + ν) where E is Young’s modulus, which is a measure of tensile stiffness. Shear modulus G is measure of elastic shear stiffness and bulk modulus k is measure of material resistance to compression. The expressions for Lamé coefficients may be inserted into Equation (218) as 2 E ν 2E SX = δijEkk + Eij ij 2 (1 + v) (1 − 2ν) 2(1 + ν) | {z } | {z } 2G 2G (221) " ν # SX = 2G Eij + δijEkk ij (1 + 2ν) Chapter 2. Finite strain theory 64

In the aim of expressing stress-strain relation compatible with FEM implementation, the following mathematical tensor operations may be considered

Ekk = δklEkl (222) and 1 Eij = [δikδjl + δilδjk] Ekl 2 1 = [δikδjlEkl + δilδjkEkl] (223) 2 1 1 = Eij + Eij = Eij 2 2 Hence, the Equation (221) can be expressed as   1 ν SXij = 2G  (δikδjl + δilδjk) Ekl + δij δklEkl 2 (1 − 2ν) | {z } | {z } "1 ν # (224) SX = 2G (δikδjl + δilδjk) + δijδkl Ekl ij 2 (1 − 2ν) | {z } Dijkl where the under-braced terms are referring to mathematical operations in Equations (222) and (223). The material tensor Dijkl is referred to as DfX and deﬁned in 2D as   Df1111 Df1122 Df1112     DfX = Df2211 Df2222 Df2212 (225)   Df1211 Df1222 Df1212 and the corresponding 3D as   Df1111 Df1122 Df1133 Df1112 Df1123 Df1113      Df2222 Df2233 Df2212 Df2223 Df2213      Df3333 Df3312 Df3323 Df3313 DfX =   (226)    Df1212 Df1223 Df1223      Sym. Df2323 Df2313   Df1313 65

Chapter 3

FEM formulation

aterial response is the basis upon which FEM solutions stand. This is the fundamental definition of engineering materials, which is first and foremost M are taken into account in designing structures. Response of materials also makes it possible to distinguish between property materials such as elastic, plastic or fracture . . . etc. Therefore, in predictive modeling of FEM based solutions, using computational tools to determine response of materials, it is important to define appropriate kinematics and employ compatible measures of stress and strain. Formulation in FEM involves approximation of mathematical expressions to discreet counterparts and construction of integral of inner product of a residual and weight functions. The trial functions are fitted into the PDE in the aim of minimizing error caused by approximation, and the weight functions are polynomial functions utilized to project the residual. Choice of weight functions may be strongly associated with nature of the problem and stability issues. The procedure eliminates spatial derivatives and allows expressing PDE:s with a set of Ordinary Differential Equation (ODE):s for transient problems and algebraic equations for steady-state problems. The approximation is referred to as Galerkin method. The advantages with FEM steams particularly from accurate repre- sentation of complex geometry, flexibility in assigning dissimilar material properties and capturing local effects through its mesh. The solution is based on numerical integration and incremental solution technique, where a linearized equations are solved consistently and progressively. In this chapter, FEM formulation as well as solution ofTL andUL for geometrically non- linear problems in 2D and 3D are presented. Solution associated with rate-independent problems (static) are presented forUL andTL. For rate-dependent (dynamic) problems aTL based solution is implemented in Newmark- β and Energy Conserving Algorithm time-integration methods are presented. The chapter concludes with numerical integration and solution technique for solving non-linear differential equations in conjunction with appropriate and stable algorithms. Chapter 3. FEM formulation 66

3.1 Principle of virtual work

In accordance to principle of virtual work, all external work are equivalent to internal work. Hence, it is possible to solve the geometric non-linear problem based on internal virtual work in an FEM approach. The virtual variation of internal work expressed in terms of Lagrangian (material) coordinate X is given in Equation (65). In which, Green- G−L Lagrange strain E is expressed as a function of second Piola-Kirchhoﬀ stress SX and the integration is taken with respect to reference frame. Z G−LT δWint = δEf SX dV0 (227) V0

G−L Green-Lagrange strain Ef expressed in terms of displacement gradient H, Equation (14), can be expressed using index notations as     ∂w ∂w ∂w ∂w  G−L 1  i j k k  Eij =  + +  (228) 2  Xj Xi ∂Xi ∂Xj  | {z } | {z } H+HT HHT where the underbraced quantities in the equation above is archived after summation G−L convention. Alternatively, the Green-Lagrange strain Ef can be computed as a tensor according to Equation (13) and reshaped in the aim of compatibility in FEM. 1 EG−L = (F T F − I) (229) f 2 G−L where I is an identity matrix. The expression for Ef can alternatively be formulated through derivative operators ∇0 and ∇w and non-linear operator A{w} as

G−L 1 E = ∇0w + A{w}∇ww (230) f 2 G−L Utilizing the expression in (230) for virtual variation in Green-Lagrange strain δEf yields G−L 1 δE = δ(∇0w) + δ(A{w}∇ww) (231) f 2 Partial diﬀerentiation with respect to virtual variation returns

G−L 1 1 δE = δ(∇0w) + δ(A{w}) ∇ww + A{w} δ(∇ww) (232) f 2 2 It is important to note that the virtual operator δ is independent on virtual variation and has commutative property such as

A{w} ∇wδw = A{δw} ∇ww (233)

The condition above allows formulation of virtual variation in Green-Lagrange strain G−L δEf in (232), as G−L δEf = ∇0δw + A{w}∇wδw (234) Chapter 3. FEM formulation 67

The external virtual variation of virtual work is given as Z Z T 0 T 0 δWext = (δw) t dS0 + (δw) ρb dV0 (235) S0 V0 where (δw) denotes virtual variation in displacement, and ρ is material density. Body force is denoted by b, and traction vector by t0. The superscript zero is used to deﬁne the initial conﬁguration.

3.1.1 Matrix formulation in 2D

In a two dimensional space, the Green-Lagrange strain tensor is expressed as a column vector assuring compatibility in a FEM approach. The vector is extracted from Green- Lagrange strain tensor accordingly.   Exx     E E G−L   G−L xx xy Ef =  Eyy  ←− Ef =   (236)   Eyx Eyy 2Exy

In the reference frame where properties of the material body are known, the coordinate 0 0 X1 correspond to x and X2 correspond to y . For a two dimensional case considering components of (236), Green-Lagrange strain may be computed as

   2 2  ∂wx ∂wx ∂wy ∂x0 ∂x0 + ∂x0     ∂wy 1 2 2 G−L    ∂wx ∂wy  Ef =  ∂y0  +  0 + 0  (237)   2  ∂y ∂y   ∂w    ∂wx y ∂w ∂w ∂wy ∂wy 0 + 0 x x ∂y ∂x 2 ∂x0 ∂y0 + 2 ∂x0 ∂y0

Alternatively, Green-Lagrange strain can be computed as a tensor according to Equation (229), and its components can be arranged according to (236), where the deformation gradient tensor is constructed as   ∂wx ∂wx ∂x0 ∂y0 F =   (238)  ∂wy ∂wy  ∂x0 ∂y0

Second Piola-Kirchhoﬀ stress SXf in a constitutive relation can be determined as       Sxx D11 D12 D13 Exx             SXf = Syy  = D21 D22 D23  Eyy  (239)       Sxy D31 D32 D33 2Exy | {z } DX where subscript f denotes FEM formulation and implies that the quantity no longer has a real tensor property. The constitutive material stiﬀness matrix DX is a strongly Chapter 3. FEM formulation 68 associated with material behaviour, such as isotropic or anisotropic.

Linear and non-linear derivative operators ∇0 and ∇w are given as

 ∂    ∂x0 0 ∂   ∂x0 0  ∂   ∂   ∂y0 0  ∇  0 0  ∇   0 =  ∂y  ; w =  ∂  (240)    0 0  ∂ ∂  ∂x  ∂y0 ∂x0  ∂  0 ∂y0 and the displacement matrix w is arranged as   wx 0 wy 0 w =   (241) 0 wx 0 wy

The external work matrix formulation in (235) in 2D is given as the following.         w δw t0 b0 w x δw x t0 x b0 x =   ; =   ; =  0 ; =  0 (242) wy δwy ty by

3.1.2 Matrix formulation in 3D

The Green-Lagrange strain tensor is expressed in terms of Lagrangian (material) coordinates X. The FEM arrangement of the strain tensor may be constructed as   Exx      Eyy      E E E   xx xy xz G−L  Ezz  G−L   E =   ←− E = E E E  (243) f   f  yx yy yz 2Exy     E E E   zx zy zz 2Eyz    2Ezx where Exy = Eyx, Eyz = Ezy and Ezx = Exz, hence, in a vector form the corresponding strains are scaled by 2. In some literature, the shear strains are arranged diﬀerently. Hence, the derivative operators as well as coordinate transformation matrix G must be arranged accordingly, review (191). It is recommended to compute Green-Lagrange strain in a 3D case based on Equation (229) to ensure computation speed, where the deformation gradient is given as   Fxx Fxy Fxz     F = Fyx Fyy Fyz (244)   Fzx Fzy Fzz Chapter 3. FEM formulation 69

Consequently, the second Piola-Kirchhoff stress SXf in a constitutive relation can be determined as       Sxx D11 D12 D13 D14 D15 D16 Exx             Syy  D21 D22 D23 D24 D25 D26  Eyy              Szz  D31 D32 D33 D34 D35 D36  Ezz  SX =   =     (245) f       Sxy D41 D42 D43 D44 D45 D46 2Exy             Syz  D51 D52 D53 D54 D55 D56 2Eyz        Szx D61 D62 D63 D64 D65 D66 2Ezx | {z } DX where the constitutive material stiffness matrix DX is defined in the reference frame and strongly dependent on material behaviour, such as isotropic or anisotropic.

Linear and non-linear derivative operators ∇0 and ∇w are given as

 ∂  0 0 0  ∂x   ∂   0 0 0     ∂y  ∂   ∂x0 0 0  ∂     0 0 0   ∂   ∂z   0 ∂y0 0   ∂     0 0 0   ∂   ∂x   0 0 ∂z0   ∂     0 0 0  ∇0 =  ∂ ∂  ; ∇w =  ∂y  (246)  0 0 0     ∂y ∂x   ∂     0 0 0   0 ∂ ∂   ∂z   ∂z ∂y   ∂     0 0 0  ∂ ∂  ∂x  0   ∂z ∂x  ∂   0 0 0   ∂y   ∂  0 0 ∂z0 and the displacement matrix is arranged as   wx 0 0 wy 0 0 wz 0 0     w =  0 wx 0 0 wy 0 0 wz 0  (247)   0 0 wx 0 0 wy 0 0 wz

Matrix formulation of external work components in Equation (235) is given as the followings.      0  0 wx δwx tx bx             0  0 0  0 w = wy ; δw = δwy ; t = ty ; b = by (248)      0  0 wz δwz tz bz Chapter 3. FEM formulation 70

3.2 Linear Shape functions

In a FEM approach, linear shape functions are used for interpolation of solution between nodal points according to given degrees-of-freedom. The study object is meshed with the desired element and appropriate properties are assigned to each element. Stresses and strains as well as stiﬀness matrix are computed for each discrete element.

(a) (b)

Figure 21 – Illustration of (a) triangular mesh and (b) tetrahedral mesh.

The element stiffness matrix is computed on the basis of numerical integration and summation over all Gauss points in the element, which is finally assembled to a global stiffness matrix. Numerical integration may be performed according to Algorithm1, and approximation Gauss points in Tables1,2. The global stiffness matrix is solved for given boundary conditions, whereby displacement at each degree-of-freedom is determined.

Ka = f =⇒ a = K−1f (249)

It must be emphasized that the stiﬀness matrix in geometrically non-linear analysis is dependent on the element displacement and non-linearity, which enters the strain displacement matrix B before integration over the material volume. Chapter 3. FEM formulation 71

3.2.1 Triangular element

Linear triangular elements utilized in two dimensional FEM approximation are characterized by an area and thickness. The element is commonly utilized in plane stress or plane strain elasticity with elastic properties, Young’s modulus E and Poisson’s ratio ν for isotropic materials, and in addition shear modulus G for orthotropic materials. The nodes are ordered in counter-clockwise and degrees of freedom at each node are given according to the element axis. y η

p3(x3, y3)

p2(x2, y2) p p (x , y ) 2 1 1 1 p x 1 ξ

Figure 22 – Illustration of 2D FEM triangular element.

The corresponding shape functions associated with each nodal point are given as 1 N e(x0, y0) = x0y0 − x0y0 + (y0 − y0)x0 + (x0 − x0)y0 1 2A0 2 3 3 2 2 3 3 2 1 N e(x0, y0) = x0y0 − x0y0 + (y0 − y0)x0 + (x0 − x0)y0 (250) 2 2A0 3 1 1 3 3 1 1 3 1 N e(x0, y0) = x0y0 − x0y0 + (y0 − y0)x0 + (x0 − x0)y0 3 2A0 1 2 2 1 1 2 2 1 where the zero superscript denotes reference conﬁguration of nodal points in Lagrangian (material) coordinates X. Area of the element is denoted by A0, which is known in the initial conﬁguration and can be calculated based on the element coordinates as

 0 0 1 x1 y1 1   A0 = det 1 x0 y0 (251) 2  2 2  0 0 1 x3 y3 It must be emphasized that the area corresponds to the Jacobian matrix J in numerical integration. Displacement w of degrees-of-freedom associated with each nodal point, sorted counter-clockwise, can be approximated by   a1     a2     e e e   0 N1 0 N2 0 N3 0 a3 w = N(x )a =     (252) e e e   0 N1 0 N2 0 N3 a4     a5   a6 where a is nodal displacement vector. Chapter 3. FEM formulation 72

3.2.2 Tetrahedral Element

For and ten node tetrahedral elements are commonly employed in three dimensional FEM analysis. The element is characterized by a volume and has elastic properties, Young’s modulus E, Poisson’s ratio ν for isotropic materials, and in addition shear modulus G for orthotropic materials. y η

p4(x4, y4, z4)

x ξ

p1(x1, y1, z1) p3(x3, y3, z3) z ζ p2(x2, y2, z2) Figure 23 – Illustration of 3D FEM tetrahedral element.

A tetrahedral element has 4 nodes and is deﬁned in an isoparametric approach, where each node has 3 degrees-of-freedom corresponding to a total of 12 degrees-of-freedom per element. This type of element is referred to constant strain tetrahedron due to the fact that strain inside the element is constant. The element has the following shape functions.

e N1 = 1 − ξ − η − ζ e N2 = ξ (253) e N3 = η e N4 = ζ Transformation between reference and spatial coordinates is done through a Jacobin matrix as  ∂N   ∂N   ∂x   ∂ξ    −T    ∂N  = J  ∂N  (254)  ∂y   ∂η   ∂N   ∂N  ∂z ∂ζ where the Jacobin matrix includes derivative of the shape functions, multiplied by known coordinates in the initial conﬁguration. Otherwise stated, the Jacobian includes purely known quantities. In an isoparametric approach, the Jacobian J is computed through multiplying derivative of the shape functions with respect to spatial axis by element Chapter 3. FEM formulation 73 coordinates as  ∂N eT ∂N eT ∂N eT  ∂ξ x ∂η x ∂ζ x    ∂N eT ∂N eT ∂N eT  J =  y y y  (255)  ∂ξ ∂η ∂ζ   ∂N eT ∂N eT ∂N eT  ∂ξ z ∂η z ∂ζ z where xe, ye and ze are row vectors including local element coordinates at each degree- of-freedom, from point 1 to point 4, Figure 23. e h i x = x1 x2 x3 x4 e h i y = y1 y2 y3 y4 (256) e h i z = z1 z2 z3 z4 Shape function N is a row matrix, including local shape functions associated with each nodal point. h e e e ei N = N1 N2 N3 N4 (257) Alternatively the Jacobian J can be simpliﬁed through coordinate transformation. The shape functions, alternatively in the reference frame can be expressed as 0 0 0 0 0 0 0 x = x1 + (x2 − x1)ξ + (x3 − x1)η + (x4 − x1)ζ 0 0 0 0 0 0 0 y = y1 + (y2 − y1)ξ + (y3 − y1)η + (y4 − y1)ζ (258) 0 0 0 0 0 0 0 z = z1 + (z2 − z1)ξ + (z3 − z1)η + (z4 − z1)ζ and the corresponding Jacobian J as   ∂x ∂x ∂x  0 0 0 0 0 0 ∂ξ ∂η ∂ζ x2 − x1 x3 − x1 x4 − x1      ∂y ∂y ∂y   0 0 0 0 0 0 J =  ∂ξ ∂η ∂ζ  ⇐⇒ J = y2 − y1 y3 − y1 y4 − y1  (259)      ∂z ∂z ∂z  0 0 0 0 0 0 z2 − z1 z3 − z1 z4 − z1 ∂ξ ∂η ∂ζ The volume of a tetrahedron can be computed on the basis of the nodal coordinates, or alternatively through the Jacobian as

 0 0 0 1 x1 y1 z1 1  0 0 0 1 V0 = det 1 x y z  ⇐⇒ V0 = det (J) (260) 6  2 2 2 6  0 0 0 1 x3 y3 z3 Local displacement w of each nodal point of a tetrahedral element can be approximated through the linear shape functions as   a1     e e e e   N1 0 0 N2 0 0 N3 0 0 N4 0 0  a2      0  e e e e   .  w = N(x )a =  0 N1 0 0 N2 0 0 N3 0 0 N4 0   .      e e e e   0 0 N1 0 0 N2 0 0 N3 0 0 N4 a11   a12 (261) where the nodal displacements are denoted by a and sorted according to degrees of freedom associated with p1 to p4, Figure 23. Chapter 3. FEM formulation 74

3.3 Strain displacement

Strain displacement matrix B and constitutive material stiffness matrix DX are the basis upon which stiffness of a material lies. This defines fundamental behaviour of the material body such as geometrically linear or non-linear .The constitutive matrix DX depends on material behaviour such as isotropic or anisotropic as well as computation framework assumption such as plane stress or plane strain. The plane stress assumption is utilized in dealing with thin elements, such as shells. Whereas plane strain is utilized in dealing with thick elements. The strain displacement matrix is integrated over volume of the material in the aim of evaluating stiffness matrix K. In classical engineering, the stiffness matrix given in Equation (1) is computed as Z T K = B DX B dV0 (262) V0 where it must be noted that the strain displacement matrix is independent on element displacement. In geometrically non-linear analysis the strain displacement matrix B is dependent on the element displacement, which must be taken into consideration. Green-Lagrange strain formulation depends on a linear virtual displacement and non linear virtual displacement, given in (234) as G−L δEf = ∇0 δw + A{w} ∇wδw (263) | {z } | {z } linear non−linear Virtual displacement δw is approximated with shape functions N(x0). It is important to note that delta (δ) has commutative property. The linear term if virtual variation of G−L Green-Lagrange strain δEf may be computed as

0 0 δw = N(x ); ∇0(δw) = ∇0 N(x ) c = B0c (264)

In a FEM approach, the displacement is approximated by shape functions, Equations G−L (252) and (261). Hence, the linear term of the Green-Lagrange strain δEf may be approximated as

0 0 w = N(x )a; ∇0w = ∇0(N(x ))a =⇒ ∇0w = B0a (265) where a is element displacement vector at current state of iteration and B0 is linear strain displacement matrix computed as

0 B0 = ∇0N(x ) (266)

G−L The non linear term in Green-Lagrange strain δEf depends on two terms, a non- linear displacement gradient matrix A{w} and a linear displacement gradient matrix

∇wδw associated with virtual variation in displacements. The non-linear displacement Chapter 3. FEM formulation 75

G−L gradient matrix A{w} in total Green-Lagrange strain i.e. Ef , Equation (230), may be approximated as the following

1 1 0 1 A{w}∇ww = A{w}∇w(N(x ))a = A{w}Hwa (267) 2 2 2 where Hw corresponds to non-linear strain displacement matrix ∇wδw and approximated as 0 Hw = ∇wδw =⇒ Hw = ∇w(N(x )) (268)

The corresponding non-linear displacement gradient in virtual variation of Green-Lagrange G−L strain δEf given in (263) can be formulated in terms of geometrically non-linear strain displacement Hw as

0 A{w}∇ww = A{w}∇w(N(x ))c = A{w}Hwc (269)

G−L where it must be emphasized that the virtual variation of Green-Lagrange strain δEf G−L enters the principle of virtual work in (227), whereas total Green-Lagrange strain Ef is utilized in determining second Piola Kirchhoﬀ stress SXf .

G−L SXf = DX Ef (270)

G−L Consequently, the non-linear term in virtual variation of Green-Lagrange strain δEf in (263) corresponds to a non-linear strain displacement matrix Bw due to the non-linear characteristics of displacement gradient matrix A{w} computed as

A{w} = ∇0w (271) and the materially non-linear strain displacement matrix is formulated as

Bw = A{w}Hw (272)

Substituting expressions for linear strain displacement matrix B0 in (266), and non-linear strain displacement matrix Bw in (272) into virtual variation of Green-Lagrange strain in (263) yield G−L δEf = (B0 + Bw)c (273)

G−L The corresponding total Green-Lagrange strain Ef in Equation (230) yield

G−L 1 E = B0 + Bw a (274) f 2 where it is noted that the displacement w approximation is given in (252) and (261). Chapter 3. FEM formulation 76

3.3.1 Matrix formulation in 2D

In a 2D approach, linear triangular shape functions are utilized in FEM approximation. For which, shape functions are outlined in section 3.2.1. The linear strain displacement matrix B0 in (266) is constructed as

 ∂N1 ∂N2 ∂N3  ∂x0 0 ∂x0 0 ∂x0 0   0  ∂N1 ∂N2 ∂N3  B0 = ∇0N(x ) =  0 ∂y0 0 ∂y0 0 ∂y0  (275)   ∂N1 ∂N1 ∂N2 ∂N2 ∂N3 ∂N3 ∂y0 ∂x0 ∂y0 ∂x0 ∂y0 ∂x0

The non-linear strain displacement matrix given in Equation (268) is constructed as   ∂N1 ∂N2 ∂N3 0 0 0 0 0 0  ∂x ∂x ∂x   ∂N1 ∂N2 ∂N3  0  ∂y0 0 ∂y0 0 ∂y0 0  Hw = ∇wN(x ) =   (276)  ∂N1 ∂N2 ∂N3   0 ∂x0 0 ∂x0 0 ∂x0    ∂N1 ∂N2 ∂N3 0 ∂y0 0 ∂y0 0 ∂y0

The non-linear displacement gradient matrix A{w} is constructed according to (277) and its components are extracted from a product of non-linear strain displacement matrix Hw multiplied by element displacement a at the current state, and reshaped accordingly.   ∂wx ∂wy 0 0 0 0  ∂x ∂x   ∂wx ∂wy  A{w} = ∇0w =  0 0 0 0  (277)  ∂y ∂y    ∂wx ∂wx ∂wy ∂wy ∂y0 ∂x0 ∂y0 ∂x0 where the components are extracted from

h iT ∂wx ∂wx ∂wy ∂wy ∂x0 ∂y0 ∂x0 ∂y0 = Hwa (278)

As previously declared, constitutive material stiffness matrix DX and strain displacement matrix B are the basis upon which stiffness of a material lies. The constitutive matrix for an isotropic material under plane strain framework is given as   1 ν 0 E   DX = ν 1 0  (279) − ν2   1  1  0 0 2 (1 − ν) where Young’s modulus E and Passion’s ratio ν are known quantities defined in the reference frame. Chapter 3. FEM formulation 77

3.3.2 Matrix formulation in 3D

In matrix formulation of strain displacements in 3D tetrahedral shape functions are employed in the aim of FEM approximation. For which, shape functions are outlined in

Section 3.2.2. The linear strain displacement matrix B0 in (266) is constructed as

 ∂N1 ∂N2 ∂N3 ∂N4  ∂x0 0 0 ∂x0 0 0 ∂x0 0 0 ∂x0 0 0    ∂N1 ∂N2 ∂N3 ∂N4   0 0 0 0 0 0 0 0 0 0 0 0   ∂y ∂y ∂y ∂y     ∂N1 ∂N2 ∂N3 ∂N4  0  0 0 ∂z0 0 0 ∂z0 0 0 ∂z0 0 0 ∂z0  B0 = ∇0N(x ) =    ∂N1 ∂N1 ∂N2 ∂N2 ∂N3 ∂N3 ∂N4 ∂N4   0 0 0 0 0 0 0 0 0 0 0 0   ∂y ∂x ∂y ∂x ∂y ∂x ∂y ∂x     ∂N1 ∂N1 ∂N2 ∂N2 ∂N3 ∂N3 ∂N4 ∂N4   0 ∂z0 ∂y0 0 ∂z0 ∂y0 0 ∂z0 ∂y0 0 ∂z0 ∂y0    ∂N1 ∂N1 ∂N2 ∂N2 ∂N3 ∂N3 ∂N4 ∂N4 ∂z0 0 ∂x0 ∂z0 0 ∂x0 ∂z0 0 ∂x0 ∂z0 0 ∂x0 (280) The non-linear strain displacement matrix given in Equation (268) is constructed as

 ∂N1 ∂N2 ∂N3 ∂N4  ∂x0 0 0 ∂x0 0 0 ∂x0 0 0 ∂x0 0 0    ∂N1 ∂N2 ∂N3 ∂N4   0 0 0 0 0 0 0 0 0 0 0 0   ∂y ∂y ∂y ∂y     ∂N1 ∂N2 ∂N3 ∂N4   ∂z0 0 0 ∂z0 0 0 ∂z0 0 0 ∂z0 0 0     ∂N1 ∂N2 ∂N3 ∂N4   0 0 0 0 0 0 0 0 0 0 0 0   ∂x ∂x ∂x ∂x    0  ∂N1 ∂N2 ∂N3 ∂N4  Hw = ∇wN(x ) =  0 ∂y0 0 0 ∂y0 0 0 ∂y0 0 0 ∂y0 0     ∂N1 ∂N2 ∂N3 ∂N4   0 0 0 0 0 0 0 0 0 0 0 0   ∂z ∂z ∂z ∂z     ∂N1 ∂N2 ∂N3 ∂N4   0 0 ∂x0 0 0 ∂x0 0 0 ∂x0 0 0 ∂x0     ∂N1 ∂N2 ∂N3 ∂N4   0 0 0 0 0 0 0 0 0 0 0 0   ∂y ∂y ∂y ∂y   ∂N1 ∂N2 ∂N3 ∂N4  0 0 ∂z0 0 0 ∂z0 0 0 ∂z0 0 0 ∂z0 (281) The non-linear displacement gradient matrix A{w} is constructed according to (277) and its components are extracted from Hwa and reshaped accordingly.   ∂wx ∂wy ∂wz ∂x0 0 0 ∂x0 0 0 ∂x0 0 0    ∂wx ∂wy ∂wz   0 ∂y0 0 0 ∂y0 0 0 ∂y0 0     ∂wx ∂wy ∂wz   0 0 0 0 0 0 0 0 0  A{w} =  ∂z ∂z ∂z  (282)  ∂wx ∂wx ∂wy ∂wy ∂wz ∂wz   0 0 0 0 0 0 0 0 0   ∂y ∂x ∂y ∂x ∂y ∂x   ∂wx ∂wx ∂wy ∂wy ∂wz ∂wz   0 0 0 0 0 0 0 0 0   ∂z ∂y ∂z ∂y ∂z ∂y  ∂wx ∂wx ∂wy ∂wy ∂wz ∂wz ∂z0 0 ∂x0 ∂z0 0 ∂x0 ∂z0 0 ∂x0 where the components are extracted from

h iT ∂wx ∂wx ∂wx ∂wy ∂wy ∂wy ∂wz ∂wz ∂wz ∂x0 ∂y0 ∂z0 ∂x0 ∂y0 ∂z0 ∂x0 ∂y0 ∂z0 = Hwa (283)

It has been highlighted that the constitutive material stiﬀness matrix DX and strain displacement matrix B play a central role in construction of material stiﬀness matrix. Chapter 3. FEM formulation 78

The constitutive matrix for an isotropic material is given as   1 − ν ν ν 0 0 0      1 − ν ν 0 0 0      E  1 − ν 0 0 0  DX =   (284)   (1 + ν)(1 − 2ν)  1 − 2ν 0 0       Sym. 1 − 2ν 0    1 − 2ν where Young’s modulus E and Passion’s ratio ν are known quantities defined in the reference frame. The compliance matrix for an orthotropic material is given as   1 νrl νtl E − E − E 0 0 0  l r t   νlr 1 νtr  − − 0 0 0   El Er Et  − νlt − νrt 1 0 0 0   El Er Et  CX =   (285)  1   0 0 0 0 0   Glr   0 0 0 0 1 0   Glt   1  0 0 0 0 0 Grt which is an inverse of the constitutive material stiffness matrix DX . Thus, the constitutive matrix for an orthotropic material is defined as

−1 DX = CX (286)

In the compliance matrix Young’s modulus E, Passion’s ratio ν and shear modulus G are given in radial (r), tangential (t) and longitudinal (l) directions. The quantities are deﬁned in the reference frame, where they are known. The given constitutive matrices comprise St.Venant-Kirchhoﬀ material, for which a generalized formulation is given in (224). Chapter 3. FEM formulation 79

3.4 Static incremental solution

In an incremental solution approach, the calculation procedure is repeated through loads steps and convergence iterations until the numerical error approaches convergence criteria. The error is denoted by the Greek letter ψ and desired to become as small as possible or approach zero. In accordance to fundamental principle of virtual work’s presumption that internal work is equal to external work a numerical solution may be formulated based on diﬀerence between internal and external work with respect to the numerical error. f ext(a) − f int(a) − ψ(a) = 0 (287) −ψ(a) = f int(a) − f ext(a) where it must be emphasized that the internal force f int is non-linear due to characteristics of non-linear strain displacement matrix Bw. A truncated Taylor’s expansion is utilized in the aim of determining a linearized residual ψ

ψ(a + da) = ψ(a) + dψ(a) (288) where dψ(a) is desired to approach zero at convergence and the minus sign of the residual is dropped for convenience. Virtual variation of internal work given in (227) can be formulated in terms of strain G−L displacement matrix corresponding to virtual variation in Green-Lagrange strain δEf , Equation (273). Z T T δWint = c (B0 + Bw) SXf dV0 (289) |{z} V0 | {z } where the under-braced terms correspond to virtual variation in Green-Lagrange strain G−L δEf and Corollary 2.2.1 is operating. Virtual variation in external work may be approximated using linear shape functions as Z Z T T 0 T T 0 δWext = c N t dS0 + c N ρb dV0 (290) S0 V0 The principle of virtual work is defined accordingly. Z Z Z T T T T 0 T T 0 c (B0 + Bw) SXf dV0 = c N t dS0 + c N ρb dV0 (291) V0 S0 V0 The final FEM formulation of principle of virtual work by definition becomes Z Z Z T T 0 T 0 (B0 + Bw) SXf dV0 = N t dS0 + N ρb dV0 (292) V0 S0 V0 | {z } | {z } f int f ext Chapter 3. FEM formulation 80

3.4.1 Total Lagrangian solution

Total Lagrangian solution is built upon relations between initial and current configuration, where desired properties of the material body through deformation, are tracked from the initial configuration. This can only be archived by employing Green-Lagrange G−L strain Ef and second Piola-Kirchhoff stress SXf . It is important to recall that the Total Lagrangian formulation is expressed in terms of Lagrangian (material) coordinates X, where all derivatives and integrals are taken with respect to reference frame of the material body. Principle of virtual work given in Equation (292) is expressed in terms of internal and external forces. External forces f ext in rate-independent (static) analysis are constant and independent on time. Hence, the critical component of the equilibrium is the internal forces f int. Differentiation of the residual in (287) yield Z T dψ(a) = df int(a) = d (B0 + Bw) SXf V0 Z Z T T (293) = dBwSXf dV0 + (B0 + Bw) dSXf dV0 V0 V0 | {z } where it must be noted that the linear strain displacement matrix B0 is constant and doesn’t depend on material displacement, whereas the non-linear strain displacement matrix Bw depends on material displacement at current state. The material displacement enters Bw through non-linear displacement gradient matrix A{w}. Thus, Bw must be updated through iterations. The under-braced term in the equation above corresponds to non-linear strain displacement. Hence, based on relation introduced in (272) can be expressed as Z Z T T dBwSXf dV0 = HwA{w}SXf dV0 (294) V0 V0 where Corollary 2.6.1 is operating as

T T dBw = d (A{w}Hw) T T = dA{w} Hw (295) T = HwA{dw}

The non-linear displacement gradient matrix A{w} is given in the same manner as in (277) for a two-dimensional case and in (277) for a three-dimensional case as

 ∂(dwx) ∂d(wy)  ∂x0 0 ∂x0 0    ∂(dwx) ∂(dwy)  A{dw} =  0 ∂y0 0 ∂y0  (296)   ∂(dwx) ∂(dwx) ∂(dwy) ∂(dwy) ∂y0 ∂x0 ∂y0 ∂x0 Chapter 3. FEM formulation 81 and

 ∂(dwx) ∂(dwy) ∂(dwz)  ∂x0 0 0 ∂x0 0 0 ∂x0 0 0    ∂(dwx) ∂(dwy) ∂(dwz)   0 ∂y0 0 0 ∂y0 0 0 ∂y0 0     ∂(dwx) ∂(dwy) ∂(dwz)   0 0 0 0 0 0 0 0 0  A{dw} =  ∂z ∂z ∂z  (297)  ∂(dwx) ∂(dwx) ∂(dwy) ∂(dwy) ∂(dwz) ∂(dwz)   0 0 0 0 0 0 0 0 0   ∂y ∂x ∂y ∂x ∂y ∂x   ∂(dwx) ∂(dwx) ∂(dwy) ∂(dwy) ∂(dwz) ∂(dwz)   0 0 0 0 0 0 0 0 0   ∂z ∂y ∂z ∂y ∂z ∂y  ∂(dwx) ∂(dwx) ∂(dwy) ∂(dwy) ∂(dwz) ∂(dwz) ∂z0 0 ∂x0 ∂z0 0 ∂x0 ∂z0 0 ∂x0 where it must be emphasized that non-linear displacement gradient matrix A{w} does not depend on type of FEM element mesh, but rather on Euclidean space dimensions. Its components are extracted in the same manner as previously from the relations below and reshaped accordingly, Equations (278) and (283). In 2D, the following relation is considered T h ∂(dwx) ∂(dwx) ∂(dwy) ∂(dwy) i ∂x0 ∂y0 ∂x0 ∂y0 = Hwda (298) and in 3D the following relation is operative

T h ∂(dwx) ∂(dwx) ∂(dwx) ∂(dwy) ∂(dwy) ∂(dwy) ∂(dwz) ∂(dwz) ∂(dwz) i ∂x0 ∂y0 ∂z0 ∂x0 ∂y0 ∂z0 ∂x0 ∂y0 ∂z0 = Hwa (299)

In the aim of compatibility with FEM and employed numerical incremental scheme, the non-linear contribution to material stiﬀness in (294) can be developed as Z Z T T HwA{dw}SXf dV0 = HwRSHw da dV0 (300) V0 V0 where RS contains second Piola-Kirchhoﬀ stress SXf components, reshaped accordingly.

In 2D, RS is constructed as   Sxx Sxy 0 0     Sxy Syy 0 0  RS =   (301)    0 0 Sxx Sxy   0 0 Sxy Syy and in 3D, RS is constructed as   Sxx Sxy Sxz 0 0 0 0 0 0     Sxy Syy Syz 0 0 0 0 0      Szy Szy Szz 0 0 0 0       0 0 0 Sxx Sxy Sxz      RS =  0 0 0 Sxy Syy  (302)      0 0 0 Szy       0 0 0 Sym.       0 0    0 Chapter 3. FEM formulation 82

Substituting the expression for non-linear contribution to the material stiﬀness in Equa- tion (300) into (293) yield Z Z T T dψ(a) = HwRSHw dV0da + (B0 + Bw) dSXf dV0 (303) V0 V0

It has been declared that second Piola-Kirchhoff stress SXf is computed according to (239) in 2D and (245) in 3D. In an incremental analysis, increment stresses depend on increment strains as G−L dSXf = DX dEf (304) G−L where increment strains dEf can be computed in the same manner as in (263) and (273) affiliated with virtual variation in strains. It is important to bear in mind that only virtual variation in strains are permitted to enter the material stiffness matrix due to the characteristics of principle of virtual work δW .

G−L dEf = ∇0dw + A∇wdw

= B0da + A∇wdwNda (305) = B0da + AHwda

= (B0 + Bw)da where ∇0dw = B0da, Hw = ∇wN and Bw = A{w}Hw. The corresponding FEM formulation of second Piola-Kirchhoﬀ stress increment dSXf becomes

dSXf = DX (B0 + Bw)da (306)

The incremental solution of the residual dψ(a) = df int(a), given in (303) can be expressed in terms of increment stresses in (306) as Z Z T T dψ(a) = HwRSHw dV0 da + (B0 + Bw) DX (B0 + Bw) dV0 da (307) V0 V0

The equation above corresponds to tangent stiﬀness matrix KT compatible with numerical incremental solution Z Z T T dψ(a) = HwRSHw dV0 + (B0 + Bw) DX (B0 + Bw) dV0 da (308) V0 V0 | {z } KT Hence, the increment solution at current state in accordance to Equation (287) becomes

−ψ(a) = f int(a) − f ext(a) (309)

Thus,

dψ(a) = KT da (310) where it is noted that only increment solution of dψ(a) is of interest considering progressive execution of the computation scheme. The internal force is given according to Equation (292) as Z T f int = (B0 + Bw) SXf dV0 (311) V0 Chapter 3. FEM formulation 83

3.4.2 Updated Lagrangian solution

Updated Lagrangian formulation is expressed in terms of Eulerian (spatial) coordinates x due to the fact that the material body has not been stationary when the computation started. This implies that the material body has already experienced kinematics between time 0 − t. The kinematics may comprise particle movement within a structural material or moisture ﬂow in case of ﬂuid dynamics. Thus, the computation starts from time t and forward, and it is natural that all derivatives and integrations must be taken with respect to the spatial frame.

The constitutive material matrix DX deﬁned in the reference frame, must be transformed to the spatial frame inUL. The constitutive matrix is given in Equations (279), (284) and (286) for 2D isotropic plane strain, isotropic and orthotropic 3D case. Transforming of the constitutive matrix is given in (88) as

−1 Dijkl = J FimFjnDmnpqFkpFlq (312)

In an incremental solution, the deformation gradient must be calculated at each load step in the scheme. Components of the deformation gradient is extracted from a product of non-linear strain displacement matrix Hw and element displacement at the current state a, Equations (298) and (299). For a two dimensional case, the components of F are extracted from T h ∂(dwx) ∂(dwx) ∂(dwy) ∂(dwy) i ∂x0 ∂y0 ∂x0 ∂y0 = Hwa (313) Likewise for a three dimensional case, the components of F are extracted from

T h ∂(dwx) ∂(dwx) ∂(dwx) ∂(dwy) ∂(dwy) ∂(dwy) ∂(dwz) ∂(dwz) ∂(dwz) i ∂x0 ∂y0 ∂z0 ∂x0 ∂y0 ∂z0 ∂x0 ∂y0 ∂z0 = Hwa (314)

The deformation gradient is deﬁned in Equation (10) as F = GRAD(w) + I, can be constructed for a 2D case and 3D case as

   ∂wx ∂wx  F11 F12 ∂x0 ∂y0 F =   =   + I (315) F F ∂wy ∂wy 21 22 ∂x0 ∂y0

    ∂wx ∂wx ∂wx F F F 0 0 0 11 12 13  ∂x ∂y ∂z     ∂wy ∂wy ∂wy  F = F F F  =   + I (316)  21 22 23  ∂x0 ∂y0 ∂z0      F31 F32 F33 ∂wz ∂wz ∂wz ∂x0 ∂y0 ∂z0 where it is noted that I is an identity matrix, deﬁned with ones in the diagonal. Mate- rial volume transformation through motion can be tracked through a so-called Jacobian matrix. For a 2D case the Jacobian matrix is given as

 ∂wx ∂wx   ∂x0 ∂y0 J = det   + I (317) ∂wy ∂wy ∂x0 ∂y0 Chapter 3. FEM formulation 84

For a 2D case, the Jacobian is extended as

 ∂wx ∂wx ∂wx   ∂x0 ∂y0 ∂z0     ∂wy ∂wy ∂wy   J = det  0 0 0  + I (318)  ∂x ∂y ∂z    ∂wz ∂wz ∂wz   ∂x0 ∂y0 ∂z0

In Updated Lagrangian approach the tangent stiffness matrix KT is computed in terms of Cauchy stress, and the constitutive matrix in spatial frame Dx. The tangent stiffness matrix is constructed based on internal energy, Equation (98), where the components are integrated over the spatial volume. Z Z ?T ? ?T ? KT = Hw RT Hwdv + B0 DxB0dv (319) v v The fact that the integration is taken with with respect to the material volume in the current state, is strongly dependent on coordinate update through the computation. It is ? important to note that the non-linear strain displacement matrix Hw is marked with a star-superscript to indicate that it gets updated at each iteration step through updating ? the Eulerian mesh. Likewise, for the linear strain displacement matrix B0. To satisfy compatibility in FEM, a matrix RT is introduced containing Cauchy stress components, which is defined in the spatial frame. The RT matrix for a 2D case is constructed as   Txx Txy 0 0     Tyx Tyy 0 0  RT =   (320)    0 0 Txx Txy   0 0 Txy Tyy

In the same manner, the RT matrix for a 3D case is constructed as   Txx Txy Txz 0 0 0 0 0 0     Txy Tyy Tyz 0 0 0 0 0      Tzy Tzy Tzz 0 0 0 0       0 0 0 Txx Txy Txz      RT =  0 0 0 Txy Tyy  (321)      0 0 0 Tzy       0 0 0 Sym.       0 0    0

Cauchy stress T defined in the current configuration, is obtained through transformation of second Piola-Kirchhoff stress SX , Equation (94). As a reminder, Piola-Kirchhoff stress

SX is defined in the initial configuration. In this regard, the second Piola-Kirchhoff stress can be computed in the same manner as in Total Lagrangian formulation as G−L 1 SX = DX E ⇐⇒ SX = DX B0 + Bw (322) f f 2 Chapter 3. FEM formulation 85 where its important to stress that the above given relation is defined in the initial configuration and non of the given matrices are updated through iteration steps. Cauchy stress T can be expressed as in (94) as

1 T T = FSX F (323) J ? The internal work f int is expressed in terms of linear strain displacement matrix B0 and Cauchy stress T . Z ?T f int = B0 T dv (324) v In the Equation above, the notation ” ? ” is used to indicate that the linear strain dis- ? placement matrix B0 is getting updated through iterations. The integration is performed over spatial volume of the material body, which can only possible through updating the Eulerian mesh through iterations. In an incremental solution, the diﬀerence between internal and external work dψ(a), also called residual, has to become as small as possible or approach zero. In Equation (287), a general expression for the residual ψ(a) is derived as

−ψ(a) = f int(a) − f ext(a) (325)

However, the residual is computed progressively in incremental solution. Therefore, in the same manner as inTL the increment residual can be computed as

dψ(a) = KT da (326) Chapter 3. FEM formulation 86

3.5 Implicit dynamic incremental solution

Dynamic analysis lays upon time-integration schemes as well as numerical solution technique schemes. The material dynamic response is dependent on the mass matrix, tangent stiffness matrix and damping. In this work time integration-schemes Newmark-β and Energy Conserving Algorithm schemes introduced in Sections 2.7.1 and 2.7.2 will play a central role in the evaluation of the predictive FEM solutions. It is also desired to employ high-speed numerical algorithms due to the fact that the time-integration scheme must be executed for small and large number of time-steps. Commonly a time length of ∆t = 10−3 and the scheme is executed for 5000 time-steps corresponding to 5 [s]. Hence, this work intends to consider aTL approach, in which material properties are defined and tracked from the reference frame. In FEM the mass matrix is determined through linear shape functions as Z T M = ρ0 N N dV0 (327) V0 where ρ0 is material density defined in the initial configuration. The damping matrix according to previously introduced Rayleigh damping model can be directly computed accordingly.

C = d1M + d2KT (328) where d1 and d2 are constants of proportionality, often determined through simulations and experiments. In this work the eﬀect of the stiﬀness will not be considered.

In time-integration schemes the residual is deﬁned as reﬀ to be distinguished from the static residual. Chapter 3. FEM formulation 87

3.5.1 Newmark-β method

In the aim of identifying a numerical solution for the load-step at state of (n + 1) the Equation of motion is expressed in terms of kinematics in the current state and the total internal energy is assigned to a residual as

reﬀ = Ma¨n+1 + Ca˙ n+1 + f int(an+1) − f ext(an+1) (329)

In the aim of ﬁnding solution at iteration state of i + 1 a truncated Taylor’s expansion is employed, where high order terms are neglected.

reff(i + 1) = reff(i) + D∆a(reff(i)) ' 0 r i r i ' = eff( ) + ∆ eff( ) 0 (330) ! ∂reff ∂reff ∂reff = reff(i) + ∆a + ∆a˙ + ∆a¨ ' 0 ∂a ∂a˙ ∂a˙ where partial derivative of the residual with respect to the kinematics follow

∂reff ∂f int ∂reff ∂reff = = KT = C = M (331) ∂a˙ ∂a ∂a˙ ∂a¨ Equations (111), (115) imply that the difference of displacements and velocities in Newmark-β approach can be expressed as 1 1 ∆a˙ = ∆a ∆a¨ = ∆a (332) β∆t β∆t2 Inserting conditions in Equations (332) and (331) into Equation (330) yield

γ γ ! reﬀ(i + 1) = reﬀ(i) + KT + C + M ∆a ' 0 β∆t β∆t2 | {z } (333)

= reﬀ(i) + Keﬀ∆a ' 0

The underbraced term in the equation above represents an effective stiffness of the system and can be simply expressed as γ γ ! Keff = KT + C + M (334) β∆t β∆t2

The given effective stiffness matrix can be expressed in terms of Newmark coefficients as

Keff = (C1 + d1C4) M + KT (335) where the second constant of proportionality d2 with respect to the tangent stiffness matrix in Rayleigh damping is not considered, Equation (102). The tangent stiffness matrix for aTL solution is given in Equation (308) as Z Z T T KT = HwRSHw dV0 + (B0 + Bw) DX (B0 + Bw) dV0 (336) V0 V0 Chapter 3. FEM formulation 88 and the internal force vector is given in Equation (311) as Z T f int = (B0 + Bw) SXf dV0 (337) V0 The difference between internal and external forces in the same manner as inTL andUL solutions, is expressed by a residual such as

?0 reﬀ = (C1 + d1C4) Man+1 + f int(an+1) − f ext(an+1) − Ma¨n (338) where ?0 ? ? a¨n = a¨n + d1a˙ n (339) and incremental solution of the system follows

Keﬀ∆ai = −reﬀ (340)

In dynamic solution, predictor steps maybe of interest in the aim of enhancing the solution scheme, such as Newton-Raphson scheme. However, the choice of predictors may not always be appropriate depending on the study case. A predictor may assume that the velocity in the upcoming load-step is approximately the same as the previous. In this regard the following predictions may be considered

a˙ n+1 = a˙ n (1 − γ) a¨n+1 = a¨n γ (341) ∆t2 an+1 = an + ∆ta˙ n + [(1 − 2β)a¨n + 2βa¨n+1] 2 A predictor may also assume that the acceleration is approximately the same as before. In this manner, the following assumptions may be considered

a¨n+1 = a¨n

a˙ n+1 = a˙ n + ∆ta¨n (342) ∆t2 an+1 = an + ∆ta˙ n + a¨n 2 It is important to emphasize that in case of utilizing kinematic predictor in the ﬁrst stage in the aim of estimating kinematics in the upcoming state, the given predictors must be corrected inside the considered scheme i.e. Newton-Raphson following

Keﬀ = ∆ai = 0 a = a + ∆a (343) a˙ = a˙ + ∆a˙ a¨ = a¨ + ∆a¨ Chapter 3. FEM formulation 89

3.5.2 Energy conserving algorithm

In the Energy Conserving Algorithm, variation in strains E? as previously referred to virtual variation in strains δE plays a key-role in formulating internal response. The rate of variation in internal forces is formulated as Z ˙ T T T Πint = a˙ BLS dV0 = a˙ f int (344) V0 and the internal force vector is formulated in Equation (157) as Z T ? ? T ? ∆a f int = (E ) S dV0 (345) V0 The internal forces comprise stress and strain variation in the system under excitation. The internal forces formulated in terms of strain energy density function given in Equation (130). The internal forces at the current state is derived in Equation (156) as

tn+1 aT f ? [Πint]tn = ∆ int (346) where the following assumptions are introduced 1 ∆E = En+1 − En; S = DfE; E = (En+1 + En) (347) 2 The variation in internal strain can be approximated as a product of strain displacement matrix and and nodal displacement vector as

? ? E = BL∆a (348)

The St. Venant-Kirchhoﬀ material assumption yielded the following relation

t 1 Z 1 Z n+1 ET DEf dV − ET DEf dV [Πint]tn = n+1 n+1 0 n n 0 2 V0 2 V0 Z (349) T = ∆E S dV0 V0 where virtual variation in strains is approximated as

?T ∆E = BL ∆a (350)

For strain variation in the system Green’s strain may be employed as ? 1 E = ∆E = B0 + A{a}Hw a (351) 2 where the displacement vector a represents nodal displacements at the current state a = an+1 − an. 1 1 ∆E = B0an+1 − B0an + an+1Hwan+1 − anHwan ∆a (352) | {z } 2 2 Chapter 3. FEM formulation 90

The underbraced term in the equation above represents linear strain displacement matrix, which can be formulated as 1 ∆E = [B0(an+1) + B0(an)] ∆a (353) 2 where the mean strain displacement matrix is given as 1 B0 = (B0(an+1) + B0(an)) (354) 2 The given relationship above demonstrates formulation of strain displacement matrix given in Equation (350). At the current state the mentioned matrix can be formulated considering impact of previous and current time step i.e. tn, tn+1. Hence, the strain displacement matrix becomes

? BL = B0 = B0(a0) (355)

Hence the mean strain displacement matrix based on mean displacement becomes

BL(a) = B0 + A{a}Hw (356) where the displacement gradient matrix A{a} is computed based on mean displacement, considering previous and current state. The displacement gradient matrix is previously referred to as A{w} to indicate that displacements are only associated with the current state, which is not the case in energy conserving. The strain displacement matrix formulated in Equation (356) allows expressing internal forces given in Equations (345) and (346) as

Z ¨ Z ¨ tn+1 ET S dV ¨aT B a S? dV ¨aT f ? [Πint]tn = ∆ 0 = ¨∆ L( ) 0 = ¨∆ int (357) V0 V0

Z ? ? f int = BL(a)S dV0 (358) V0 It is important to note that the energy conserving algorithm is derived assuming St. Venant Kirchhoﬀ material. If an arbitrary material is desired, the internal force between time tn, tn+1 must satisfy the given equality in Equation (357) such that

? 2 h T i S = S + k∆Ek wn+1 − wn − ∆E E ∆E (359) where magnitude of virtual variation is

k∆Ek2 = ∆ET ∆E (360) and mean stress is

∂w S = (361) ∂E E=E Chapter 3. FEM formulation 91

For St. Venant Kirchhoﬀ material, the following relation must be archived

S? = S = DE (362)

The above relation yield the following relation, which satisﬁes the equality in Equation 357. T ? ∆E S = wn+1 − wn (363)

In an iterative solution using truncated Taylor-series expansion the nodal displacements in the system can be computed as

Keﬀ∆ai = reﬀ (364) and the displacement associated with previous state must be added to the current state as

ai+1 = ai + ∆ai (365) where the effective stiffness matrix is derived from the residual given in Equation (150), which corresponds to reff.

2 ? D∆a(reff) = 2 M∆ai + 2D∆a(f int) (366) ∆t | {z } where the underbraced term, corresponding to effect of internal forces, can be formulated as Z Z ? T ? T ? 2D∆a(f int) = 2 BL(a) D∆a(S ) dV0 + 2 D∆a(BL(a) )S dV0 (367) V0 V0 | {z } | {z } First term Second term where the derivative operator D∆a is acting on the integral based on chain rule. In the first term of the equation above, the derivative operator acting on the strains correspond to ? ? 1 ∂S D∆a(S ) = DD∆aE −→ D = 2 (368) 2 ∂E and the second term where the derivative operator is acting on the strain displacement matrix correspond to

T 1 D∆a(BL(a) ) = D∆a(B0 + A(a)Hw) = A(a)Hw (369) 2 Hence, the entire second term in Equation (367) can be expressed as

T ? 1 T T ? 1 T ? D∆a(BL(a) )S = H A(∆ai) S = H R Hw∆ai (370) 2 w 2 w where the R? matrix includes reshaped square matrix of stresses as in staticTL andUL solutions.   S? 0 0   ?  ?  R =  0 S 0  (371)   0 0 S? Chapter 3. FEM formulation 92

The above conditions imply that the total internal forces can be expressed in Equation (367) can be expressed in its FEM formulation as Z Z ? T T ? 2D∆a(f int) = BL(a) DBL(a) dV0 + HwR Hw dV0 ∆ai (372) V0 V0 The above condition reveals the fact that the tangent stiﬀness matrix must be deﬁned as Z Z T T ? KT = BL(a) DBL(a) dV0 + HwR Hw dV0 (373) V0 V0

Equation (150) implies that the effective stiffness matrix must become 4 Keff = M + KT (374) ∆t2 The mass matrix is defined as in previous section as Z T M = ρ0 N N dV0 (375) V0 In the aim of enhancing the solution scheme, such as Newton-Raphson, kinematic predictors may be of interest. If the acceleration in the upcoming load-step is assumed to be the same as the previous one, the following predictors may be considered.

a¨n+1 = a¨n 2 a˙ n+1 = (an+1 − an) − a˙ n (376) ∆t

an+1 = an + ∆ta˙ n

It can be observed that in this integration scheme, damping is not introduced due to the fact that this scheme is utilized in veriﬁcation of Newmark-β method. If Energy Conserving Algorithm with damping is of interest, the reader may refer to [23] and [4]. Chapter 3. FEM formulation 93

3.6 Numerical integration

In a FEM analysis, a spatial domain is often meshed with hundreds of FEM elements, which in many cases are orientated and shaped differently, i.e. unstructured mesh. The individual mesh elements are analyzed separately and combined to predict response of the structure, i.e. evaluating the solution. This procedure results in complications, since FEM analysis is commonly conducted in a 2D and 3D space, and analytical (symbolic) computation of FEM equations may become too complicated. Primarily because properties of each element is often dependent on coordinates. In an incremental solution the computation is performed back and forth until equilibrium is satisfied. Performing symbolic calculation in Matlab or Python is rather too complicated and time consuming. In the aim of establishing an efficient FEM solution, numerical integration becomes significant. This plays a key role in enhancing and astonishingly improving the solution. Consider a one dimensional function f(x) to be integrated numerically. The function is given as Z Z 1 f(x) dL = f(x(ξ)) detJ dξ (377) L −1 where (detJ) is area under the function to be determined. The integral above simply comprises area under the function of coordinate limits h−1, 1i. Hence, a method of so called, quadrature formulae is desired to determine the area. In performing numerical integration, some points in the integration interval −1 ≤ ξ ≤ 1 maybe chosen as integration points and coordinate transformation from spatial coordinates to reference coordinates must be considered. This allows expressing the symbolic integral in a numerical manner as n Z X f(x) dL ' f(ξi) detJHi + R (378) L i=1 where n denotes number of integration points and Hi is weight of each integration point and R is so called reminder.

f(ξ) f(ξ)

R 1 P2 I = f(ξ) dξ I = i=1 f(ξ) Hi −1

ξ ξ −1 1 ξ1 ξ2

H1 H2

(a) (b) Figure 24 – (a) Analytical integration and (b) numerical integration. Chapter 3. FEM formulation 94

It is desired to make R approach zero, in the aim of expressing the numerical integration as n Z X f(x) dL ' f(ξi) detJHi (379) L i=1 where ξi is coordinate of the chosen position of the integration point, also called Gauss point. In the same manner, the declared numerical approach can be extended to multi- dimensional cases. For a 2D case the integral becomes Z Z 1 Z 1 f(x, y) dA = f (x(ξ, η), y(ξ, η)) detJdξdη A −1 −1 m n (380) X X ' f (x(ξi, ηj), y(ξi, ηj)) detJHiHj j=1 i=1 and likewise, for a 3D case the numerical integration is performed as Z Z 1 Z 1 Z 1 f(x, y) dV = f (x(ξ, η, ζ), y(ξ, η, ζ)) detJdξdηdζ V −1 −1 −1 p m n (381) X X X ' f (x(ξi, ηj, ζk), y(ξi, ηj, ζk)) detJHiHjHk k=1 j=1 i=1 It is important to note that the quantity of determinant of the Jacobian matrix det(J) returns, must be compatible with dimension of the equation. Hence, the Jacobian matrix is constructed such that it returns length, area and volume under a function depending on dimension of the function. In this work, numerical integration plays a central roll in evaluating FEM equations. In a 2D approach, constant strain triangular elements are chosen with the following Gauss points.

Table 1 – Triangle Gauss points.

ξi ηi w 1 1 1 6 6 3 1 2 1 6 3 3 2 1 1 3 6 3

In a 3D approach constant strain tetrahedral elements are chosen with 4 Gauss points.

Table 2 – Tetrahedral Gauss points.

ξi ηi ζi w 1 0.5854101966249685 0.1381966011250105 0.1381966011250105 24 1 0.1381966011250105 0.1381966011250105 0.1381966011250105 24 1 0.1381966011250105 0.1381966011250105 0.5854101966249685 24 1 0.1381966011250105 0.5854101966249685 0.1381966011250105 24 Chapter 3. FEM formulation 95

In the tables above, the weight function w corresponds to HiHjHk at each integration step. The numerical integration of a material stiﬀness matrix Ke may be performed according to Algorithm1.

Algorithm 1 Numerical integration. •Initialize stiffness matrix Ke •Initialize Gauss points ξi, ηi, ζi •Initialize corresponding weights w for all Number of Gauss points do Update shape functions N, B Update the Jacobian J e New stiffness matrix Ki e Pn e K = i=1 Ki . Element stiffness end for

In the Algorithm1, the matrix N represents FEM shape functions and B is gradient of the shape functions ∇N, i.e. Strain displacement matrix, [25]. Chapter 3. FEM formulation 96

3.7 Numerical solution technique

Large deformation analysis involves incremental solution in time-steps. The time increment starts from zero when the desired object is at rest, stationed in a Cartesian coordinate system, and moves ahead as the object experiences large displacement, large strain and rotation. Each time increment, also called a load step, correspond to an equilibrium position obtained through a series of iterations bounded by convergence criteria. In order to approach the deformed configuration of the object, the calculation procedure is repeated through load steps and convergence iterations until the difference between internal and external forces are acceptable. The difference is denoted by Greek letter ψ and desired to become as small as possible or approach zero.

t slope KT t+∆t slope Kt+∆t ψt ψ T Load steps

t+∆t f ext

t f ext ∆a1 ∆a2

at at+∆t Displacement

Figure 25 – Schematic illustration of Newton-Raphson scheme.

In numerical analysis, a collection of different schemes with different characteristics are available. The choice of numerical method is often associated with accuracy differences and computation speed. Traditionally, Euler forward and backward scheme or Runga- Kutta methods are utilized as an iterative computational tool. However, these methods approach convergence at small time steps, therefore longer time steps are often considered. This demands considerable time and computer power for convergence. Contrastingly, Newton-Raphson scheme has demonstrated accuracy in computation as well as converge at considerable shorter time steps. In this method the non-linear differential equations must be linearized with corresponding linear equations and solved progressively until convergence. In FEM, the equations are linearized and expressed as a tangent material stiffness, internal and external force vectors. The linearized tangent stiffness matrix is solved consistently and progressively in a Newton-Raphson scheme until difference between internal and Chapter 3. FEM formulation 97 external forces have approached convergence. The application of Newton-Raphson scheme is explained in Algorithm2.

Algorithm 2 Newton-Raphson scheme. Initialize quantities for all Load steps do Update boundary conditions Set r = f int − f ext while ψ ≥ conv. do . Covergence criteria Solve incremental displacement for all Elements do Update global matrices end for Update r Update ψ = norm(r) end while . Repeat until convergence end for

In this work, the FEM incremental solutions for bothTL andUL are concluded with a tangent stiﬀness matrix KT and an internal force vector f int in aim of ensuring compatibility with Newton-Raphson scheme. 98

Chapter 4

Implementation

arge deformation involves analysis of discretized non-linear differential equations. The discretization procedure concerns transforming equations and functions to L discrete counterparts in the aim of numerical implementation. The solution concerns corresponding linearized equations based on an iterative process bounded by convergence criteria. To acquire numerical accuracy, the difference between right and left hand-side of the equation of motion must approach zero. Hence, a numerical iteration scheme must be employed to satisfy the desired convergence criteria. In this work, the mentioned criteria is set to 1 [µN], i.e. one micro-newton and the Newton-Raphson scheme is executed for time-step length corresponding to ∆t = 1/1000, i.e. one mil- lisecond. In the first phase of the implementation in plane strain, a cantilever is chosen with material properties of steel. In static analysis aTL andUL approach is considered and executed inside a Newton-Raphson scheme. In dynamic analysis, Newmark-β and Energy Conserving Algorithm direct time-integration methods are considered. The mentioned approaches are executed inside a Newton-Raphson scheme and the results are compared with corresponding linear (engineering) solution. The comparison is conducted to determine appropriate and compatible methods to proceed with in dynamic analysis of wood in 3D. The second phase of the implementation comprises experimental validation of a built up numerical method to predict static and dynamic deformation of orthotropic wood (pine). The numerical method consists of aTL based Newmark- β time-integration method and Newton-Raphson scheme. This chapter presents numerical implementation ofTL,UL, Newmark- β and Energy Conserving Algorithm direct time-integration methods. For the mentioned approaches, step-by-step algorithms in conjunction with solution technique in Newton-Raphson are presented. The chapter concludes with experimental implementation strategy and post- processing of the experimental data. Chapter 4. Implementation 99

4.1 Geometry and material properties

4.1.1 Steel cantilever in plane strain

Geometry of materials has a signiﬁcant impact on material behaviour in large deformation analysis. In order to allow large deformation, the material must behave in accordance to constitutive model behaviour introduced previously in Equation (224). The corresponding FEM formulation of the constitutive material matrix in plane-strain is given in Equation (279), in which the material is represented by Young’s modulus of E and Poisson’s ratio of ν due to isotropic behaviour of steel. The geometry and corresponding material properties is presented in Table3. The domain mesh comprises constant strain triangular elements introduced in Section 3.2.1, and the cantilever is subjected to a vertical deformation at its free end.

Table 3 – Geometry and material properties of steel

Material Length [mm] Height [mm] Deformation [mm] E [MPa] ν Steel 500 20 200 210 · 103 0.3

y 0[mm] 20 x [mm] 200

500 [mm]

Figure 26 – Illustration of cantilever geometry in 2D.

Figure 27 – FEM model of steel cantilever in 2D with triangular mesh elements. Chapter 4. Implementation 100

4.1.2 Pine wood board cantilever in 3D

In numerical modeling of pine wood boards, the numerical geometry is governed by experimental geometry. The wood board is of knot-free softwood pine. The governing FEM material constitutive behaviour corresponds to elastic orthotropic St.Venant-Kirchhoﬀ material, in which the material is represented by Young’s modulus of E and Poisson’s ratio of ν and shear modulus G in radial, tangential and axial directions corresponding to 9 parameters, Equation (286). The given material properties are extracted from [18] and summarised in Table4. The geometry is chosen considering geometrically non-linear large deformation compatibility. The geometrical data is presented in Table5.

4 Table 4 – Material properties of pine (Eij and Gij in 10 [MPa])

Material El Er Et Glt Glr Gtr vrt vlr vtl Pine 16.40 1.130 0.90 0.91 1.18 0.079 0.063 0.043 0.024

Table 5 – Pine board’s geometrical data

Length [mm] Clamped length [mm] Width [mm] Height [mm] 1500 200 ∼ 40 8

Mechanical clamp

z 200 [mm] x

1300 [mm] [mm] 8 0 [mm] 200

~40 [mm]

Figure 28 – Illustration of experimental model of the wood board. Chapter 4. Implementation 101

The board is meshed using constant strain tetrahedral elements introduced in Section 3.2.2 and a vertical deformation at its free end.

Figure 29 – FEM model of the wood board with tetrahedral mesh elements. Chapter 4. Implementation 102

4.2 Numerical implementation

Implementation of finite deformation includes non-linear partial differential equations, which demands an incremental solution. The corresponding linearized equations of motion as well as FEM formulations and Newton-Raphson solution technique have been introduced in the previous chapter. It is important to recall that inTL formulation all the desired material quantities are tracked from the initial configuration, employing Lagrangian mesh. In this respect, the boundary nodes coincide with material particles. Thus, it is not necessary to update the mesh through iterations. Whereas, inUL formulation the desired material quantities are tracked from a previous time step, employing Eulerian mesh where boundary nodes do not coincide with material particles, Figure9. Hence, the mesh coordinates and material constitutive matrix must be updated continuously through load-steps and iterations to determine the deformed quantities of the material body. It must be emphasized that in this instance, properties of each FEM element is time dependent and varies through iterations. Therefore, all the coordinates and constitutive material properties Dx must be updated continuously and progressively through iterations. In analysis of orthotropic materials, such as wood, coordinate transformation must be considered. Hence, the constitutive matrix DX associated with each FEM must be also transformed, Section 2.8.1. In large deformation analysis, employing appropriate stress and strain measure must be given as special attention for compatibility reasons. As previously declared, in this work G−L Green-Lagrange strain E and second Piola-Kirchhoff stress SX will be employed in TL analysis. Whereas, inUL analysis, Almansi strain EA and Cauchy stress T will be operated. The calculation procedure is carried out through applying displacement increments through load steps, for which series of convergence iterations are performed until the desired convergence criteria is met. In static analysis theTL andUL and linear analysis is executed inside a Newton-Raphson scheme for 20 load steps. The corresponding kinematics and internal forces at the deformed configuration are utilized as initial conditions for the dynamic analysis. To apprehend the dynamic behaviour thoroughly a time-step length of ∆t = 1/1000 is considered. In plane strain analysis the dynamic scheme is executed for 2.5 [s] corresponding to 2500 time-steps, where physical material damping is disregarded to apprehend undamped- vibration of the system. Furthermore, a comparison of Newmark-β and Energy Conser- vation Algorithm time-integration methods are conducted to ensure numerical accuracy of Newmark-β method. In 3D analysis of pine wood board, the dynamic scheme is executed for 8 [s] corresponding to 8000 time-steps, where the material damping is modelled according Rayleigh-damping dependent on the material mass. Chapter 4. Implementation 103

To archive a structured scripting, a collection of functions are constructed, for which manuals are provided in AppendixA. Each function takes a set of inputs and provides a set of outputs. These functions are implemented inside a Newton-Raphson scheme according to the algorithms presented below. Concerning a classical engineering linear solution, the scheme is executed ignoring non-linear contribution to the material stiﬀness as well as internal forces.

Algorithm 3 Total & Updated Lagrangian in Newton-Raphson. –Generate geometry, mesh, boundary conditions –Initialize internal and external quantities

for all Elements do –Stresses and strains –Tangent Stiﬀness matrix KT end for

for all Load steps do –Update boundary conditions –Set ψ = f int − f ext while kψk ≥ 10−6 do Solve KT ai = −ψ . Increment displacement –Update kinematics for all Elements do –Generate Stresses and strains –Generate internal forces –Update material properties . In UL –Tangent Stiﬀness matrix KT end for

ψ = f int − f ext end while . Repeat until convergence –Accept quantities end for Chapter 4. Implementation 104

Algorithm 4 Newmark-β time-integration in Newton-Raphson. –Generate geometry, mesh, boundary conditions –Generate Newmark constants –Initialize internal and external quantities for all Load steps do –Update boundary conditions –New load level f int(n + 1) = f ext(tn+1) –Initiate iteration quantities a0 = a(n); S0 = S(n) ? a¨(n) = C1a(n) + C2a˙ (n) + C3a¨(n) ? a˙ (n) = C4a(n) + C5a˙ (n) 0 ? ? a¨(n) = a¨(n) + d1a˙ (n) –Predictors −6 while kreffk ≥ 10 do –Effective stiffness matrix Keff –Solve Keff ai = −reff . Increment displacement –Update kinematics ai+1 = ai−1 + ai ? aï+1 = C1ai+1 + a¨(n) ? a˙ i+1 = C4ai+1 + a˙ (n) for all Elements do –Generate Stresses and strains –Generate internal forces end for 0 reff = (C1 + d2C4)Mai+1 + f int(ai+1) − f ext(ai+1) − Ma¨(n) end while . Repeat until convergence –Accept quantities a(n+1) = ai+1; a˙ (n+1) = a˙ i+1; a¨(n+1) = aï+1; S(n+1) = Si+1 end for Chapter 4. Implementation 105

Algorithm 5 Energy Conserving Algorithm in Newton-Raphson. –Generate geometry, mesh, boundary conditions –Generate Newmark constants –Initialize internal and external quantities for all Load steps do –Update boundary conditions –New load level f int(n + 1) = f ext(tn+1) –Initiate iteration quantities a0 = a(n); S0 = S(n) –Predictors −6 while kreﬀk ≥ 10 do

–Effective stiffness matrix Keff –Solve Keff ai = −reff . Increment displacement –Update kinematics an+1 = an+1 + ai 2 a˙ i+1 = ∆t (an+1 − an) − a˙ n 2 a¨n+1 = ∆t (a˙ n+1 − a˙ n) − a¨n for all Elements do –Generate Stresses and strains –Generate internal forces end for 4 ? m 4 reff = ∆t2 M(an+1 − an) + 2f int − 2f ext − ∆t2 Ma˙ n end while . Repeat until convergence –Accept quantities a(n+1) = ai+1; a˙ (n+1) = a˙ i+1; a¨(n+1) = aï+1; S(n+1) = Si+1 end for Chapter 4. Implementation 106

4.3 Experimental implementation

The wood (pine) board is sawn and shaved to the desired size and shape given in Table5. In order to archive the mechanical clamp shown in Figure 28, the wood board is placed on a steel block clamped with F-clamps to a heavy table. On the top a splicable steel block is fastened with M12 bolts to the bottom block providing clamped boundary conditions.

Figure 30 – Experimental setup of mechanical clamp.

To measure the mechanical deformation, grid paper is placed at the end of the clamped wood board, where the mechanical displacement is applied. On its opposite side a high- speed camera is stationed to capture the dynamic oscillations, Figure 31. Accelerometers are attached to the wood board and numbered accordingly. The accelerometers only measure acceleration in the vertical direction. In dynamic analysis, the cantilever is deformed 200 [mm] and resealed to undergo dynamic motion. The accelerations are measured following the board through ﬁnite deformation, due to the fact that the accelerometers are attached to the board. Hence a transformation with respect to angle θ, Figure 31 must be conducted to determine the vertical acceleration. Further, the signal must processed appropriately. Chapter 4. Implementation 107

1280/3 [mm] 1280/3 [mm] 1280/3 [mm] 20 [mm]

3 2 1 Accelerometer

θ a

High- speed Camera Figure 31 – Experiment design strategy.

4.4 Analog ﬁlter design and analysis

The accelerometers measure acceleration of motion as the cantilever oscillates. The data comprise a signal and must be processed and transformed to determine desired quantities at given frequencies. The extracted acceleration data is given in [m/s2] referring to a fixed coordinate system. Thus it must be transformed with respect to angle θ, Figure 31. The data must be further detrended to remove both constant and linear trends before further processing. In conjunction with the experiment a modal analysis is conducted to determine the lowest elastic eigenfrequencies, with respect of which an analog filter must be designed. Hence, a high pass Butterworth filter is employed to permit passing signals with higher frequency than a cutoff frequency, and attenuate all frequencies lower than the cutoff frequency. The cutoff frequency is constructed according to Nyquist–Shannon sampling theorem, [22], to determine discrete sequence of samples and extract finite information from the accelerometers signal accordingly.

2fc ωcf = (382) fs where fc is sampling rate (frequency) of the signal in [Hz] and fs is the lowest signal al- lowed to enter the cutoﬀ frequency set to 2.5 [Hz] in this work. The acceleration data can be further utilized to determine its corresponding velocities and displacements through cumulative trapezoidal numerical integration, where the mentioned quantities are determined accordingly. Z a˙ = a¨ dt (383)

Z a = a˙ dt (384) 108

Chapter 5

Numerical results in plane strain

he outcome of the numerical analysis in plane strain is presented in this chapter. The analysis is conducted for material properties and in accordance to numer- T ical algorithms proposed in the previous chapter. In the static analysis with rate-independency a comparison ofTL andUL is conducted with respect to number of mesh elements for contraction of the cantilever through ﬁnite deformation. In the mentioned comparison, a corresponding linear solution is also included. Further, for a mesh element size of 5 [mm] TL andUL in comparison to linear (engineering) solution are compared with respect to kinematics, stresses, strains, internal and external quantities. In implicit dynamic analysis with material rate-dependency, aTL approach with mesh element size of 10 [mm] is considered to enhance the time-integration scheme with respect to computation speed. Due to uncertainties in direct time-integration methods, a linear and non-linear Newmark-β and non-linear Energy Conserving Algorithm methods are compared. The mentioned comparison validates numerical accuracy of Newmark-β time-integration method. Further dynamic analysis of kinematics and internal quantities are conducted whereTL andUL are compared with linear (engineering) solution. The mentioned approaches are implemented according to Newmark-β time-integration method with a time-length of ∆t = 1/1000 disregarding material physical damping. The analysis is executed for 2.5 [s] corresponding to 2500 time-steps and the results are presented for the mesh element, on which the mechanical displacement is applied, Figure 26. Chapter 5. Numerical results in plane strain 109

5.1 Static results in 2D

The geometrical and material non-linear large deformation analysis inTL andUL demonstrate contraction of the cantilever model through ﬁnite deformation, whereas in linear engineering solution the cantilever’s length remain the same after deformation. The TL andUL yield identical solution despite diﬀerences in their formulation and stress-strain measure.

Figure 32 – Non-linear static deformation inTL andUL with colormap of axial stresses in [MPa].

Figure 33 – Linear static deformation with colormap of axial stresses in [MPa]. Chapter 5. Numerical results in plane strain 110

A comparison of horizontal nodal displacement (contraction) of the cantilever, Figure, 26 through finite deformation inTL andUL, demonstrate identical results for different number of mesh elements as the cantilever contracts horizontally through finite deformation. However, the Engineering solution only displays nodal rotation.

20 EL. Eng. 20 EL. TL 20 EL. UL 100 EL. Eng. 100 EL. TL 100 EL. UL >800 EL. Eng. >800 EL. TL >800 EL. UL

-20

-40 Displacement [mm] 2 4 6 8 10 12 14 16 18 20 Load step [-]

Figure 34 – Comparison of horizontal displacement inTL,UL vs. linear solution.

Nodal displacements against corresponding reaction forces display a diverse solution between geometrically non-linear and linear solutions. The horizontal displacements and reaction forces are heading in diﬀerent directions due to the fact that in linear solution overall length of the cantilever remain the same after deformation. However, only nodal rotation occurs which yields the horizontal displacement. The vertical reaction forces corresponding to the applied displacement in a linear solution yields a diﬀerence in prediction corresponding to approx. 30% in comparison toTL andUL.

Linear TL UL Linear TL UL 10-10

5 -2 4 -4 3

2 -6 1 -8 0 Force [kN] Force [N]

-1 -10 -2 -12 -3

-4 -14

-40 -35 -30 -25 -20 -15 -10 -5 0 5 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 Displacement [mm] Displacement [mm] (a) (b)

Figure 35 – (a) horizontal and (b) vertical static reaction forces in plane strain.

Material response in terms of internal forces variation corresponding to the applied displacement demonstrate similar path inTL andUL solution with diﬀerence in magnitude. Chapter 5. Numerical results in plane strain 111

However, a fundamental diﬀerence in horizontal response of the linear solution. Likewise, the vertical response appear to underestimate the ﬁnal internal force by approx. 100 [N].

Linear TL UL Linear TL UL 10-9 -100 1.5 -200 1 -300 0.5 -400 0 -500 -0.5 Force [N] Force [N] -600 -1 -700 -1.5 -800 -2 -900 -40 -35 -30 -25 -20 -15 -10 -5 0 5 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 Displacement [mm] Displacement [mm] (a) (b)

Figure 36 – (a) horizontal and (b) vertical static internal forces in plane strain.

Element strains corresponding to the imposed displacement, display a diverse results between linear and non-linearTL andUL solutions. The Engineering strain develops linearly, unlike Green-Lagrange strain EG−L and Almansi EA strain, which demonstrate non-linearity through load steps. The Green-Lagrange strain and Almansi strain are heading in opposite directions althoughTL andUL approach similar solution.

10-4 (a) 10-4 (b) 10

8 -5 6 -10 4

2 -15

10 20 10 20

EG-L EA

Strain [-] 10-3 (c) 0 -0.5 -1 -1.5 -2

10 20 Load-step [-]

Figure 37 – (a) axial (b) transversal and (c) shear strains in plane strain (static). Chapter 5. Numerical results in plane strain 112

Axial normal stresses show an exponential rise in second Piola-Kirchhoﬀ stress Sxx, while

Cauchy stress Txx and Kirchhoff stress Txx increase gradually and tend to diminish rapidly after load-step 10. The axial Engineering stress measure σxx remains non-existent through all load steps, since in a linear solution the structure does not contract. Transversal normal stresses display a rapid diminish in second Piola-Kirchhoff stress Syy and Engineering stress σyy. Although while Cauchy stress T and Kirchhoff stress Tyy follows a similar path, after load-step 18 the stress curves tend to flatten.

(a) (b)

60 -100

-300 -40 10 20 10 20 S T

-100

10 20 Load-step [-]

Figure 38 – (a) axial (b) transversal and (c) shear stresses in plane strain (static).

The bi-axial shear stresses demonstrate gradual increase in second Piola-Kirchhoﬀ stress

Sxy and slight variation in the linear stress σxy, while a swifter decline in Cauchy stress

Txy following Kirchhoff stress Txy. The comparison of stresses and strains reveals a pattern betweenTL’s Green-Lagrange G−L strain E and second Piola-Kirchhoff stress SX , likewise withUL’s Almansi strain EA, Cauchy stress T and Kirchhoff stress T . This pattern confirms the compatibility of chosen stress and strain measures. The same rule can be applied to Engineering stress and strain measures. Chapter 5. Numerical results in plane strain 113

5.2 Implicit dynamic results in plane strain

Large deformation under dynamic motion inTL andUL demonstrate identical results with horizontal contraction of the cantilever through motion. Contrastingly, in linear (engineering) solution the overall length of the cantilever remains the same through motion, which lies in the inﬁnitesimal kinematics assumption.

Figure 39 – Dynamic motion ofTL andUL solution in Newmark- β scheme with colormap of axial stresses in [MPa].

Figure 40 – Dynamic motion of linear engineering solution in Newmark-β scheme with colormap of axial stresses in [MPa]. Chapter 5. Numerical results in plane strain 114

In the aim of ensuring numerical accuracy, a comparison of linear and non-linear Newmark- β and Energy Conserving Algorithm is conducted. The linear Newmark-β implies linear data steaming from linear static engineering analysis. Likewise, non-linear Newmark-β and Energy Conserving Algorithm data steaming from a static Total Lagrangian Analysis. The choice ofTL is associated with previously conducted comparison shown in Figure 34.

Linear Newmark- non-linear Newmark- Non-linear Eng. conserv.

(a) 200

100

-100

-200 0 1 2.5 (b) 20

0 Displacement [mm]

-20

-40

-60 0 1 2.5 Time [s]

Figure 41 – (a) Vertical (b) horizontal displacement of Linear and non-linear Newmark-β method vs. non-linear Energy Conserving Algorithm.

Figure 41 displays similar motion associated with non linear Newmark-β and Energy Conserving Algorithm in vertical and horizontal degrees-of-freedom. However a deviation of linear solution in the horizontal displacement can be observed. This is due to the fact that the linear solution does not consider geometrical non-linearity. Hence, the cantilever does not contract inwards through deformation, Figures 39, 40. The fact that a Linear Energy Conserving Algorithm is not included in the comparison is due to the fact that Energy Conserving Algorithm is built upon dependency of strain displacement matrix

BL(a) on the element displacement a, which is not the case for a linear solution. Chapter 5. Numerical results in plane strain 115

The dynamic response of the cantilever without physical material damping, demonstrates similar motion when non-linearTL andUL are employed. The vertical motion is almost similar due to the applied displacement as boundary condition. However, the horizontal motion displays variation in length of the cantilever through motion in non-linearTL andUL. Contrastingly, engineering solution only demonstrates horizontal rotation of the nodal points while retaining the overall length.

Linear TL UL Linear TL UL

10 200

100 -10

-20 0

-30 Displacement [mm] Displacement [mm] -40 -100

-50

-60 -200 0 1 2.5 0 1 2.5 Time [s] Time [s] (a) (b)

Figure 42 – (a) horizontal and (b) vertical dynamic displacement in plane strain.

The horizontal and vertical velocities display matching results in non-linearTL andUL. However, linear engineering solution demonstrates notably small velocity due to the fact that the horizontal motion of the cantilever in a linear solution only comprises nodal rotations. The vertical velocities displays matching results forTL andUL with small deviation in engineering solution.

Linear TL UL Linear TL UL

0 0 Velocity [m/s] Velocity [m/s]

0 1 2.5 0 1 2.5 Time [s] Time [s] (a) (b)

Figure 43 – (a) horizontal and (b) vertical velocity response in plane strain. Chapter 5. Numerical results in plane strain 116

The horizontal and vertical accelerations yield again similar solution inTL andUL. However, linear engineering solution displays a great deviation.

Linear TL UL Linear TL UL

400 1000

300 800

600

200 ] ] 2 2

100 400

0 200 Acceleration [m/s Acceleration [m/s -100 0

-200 -200

-300 -400 0 1 2.5 0 1 2.5 Time [s] Time [s] (a) (b)

Figure 44 – (a) horizontal and (b) vertical acceleration response in plane strain.

The dynamic analysis in this work has been conducted for small time steps in the aim of analyzing behaviour of the cantilever thoroughly. Hence, the time step is set to ∆t = 10−3 and the computation scheme is executed for 2500 load-steps. In this manner, speed-efficiency of each formulation becomes important. TheTL formulation has been substantially more efficient to execute in Newmark-β time-integration method and Newton-Raphson scheme than,UL with a calculation-speed approximately matching the engineering solution. However,UL has been computationally quadruple more expensive and time-consuming thanTL. The internal force variation through motion follows the same pattern asTL andUL yield matching results while notable difference in linear solution.

Linear TL UL Linear TL UL 40 30

20 20

10 0

0 -20

-10 Force [N] Force [N] -40 -20

-60 -30

-40 -80 0 1 2.5 0 1 2.5 Time [s] Time [s] (a) (b)

Figure 45 – (a) horizontal and (b) vertical dynamic internal forces in plane strain. Chapter 5. Numerical results in plane strain 117

The strains under dynamic motion display an initial strain at start due to the fact that the dynamic analysis is started at ﬁnal deformed conﬁguration of theTL static analysis. Thus, the corresponding displacements and internal forces have entered the equation of motion in the Newmark-β time-integration method as initial conditions. The axial and transversal strains inTL andUL display dissimilar solution. However, the shear strains appear to yield similar solution.

10-5 (a) 10-5 (b) 2 4 0 2 -2 0 -4 -2 0 1 2.5 0 1 2.5

EG-L EA

Strain [-] 10-4 (c) 1

-1

0 1 2.5 Time [s]

Figure 46 – (a) axial (b) transversal and (c) shear strains in plane strain (dynamic).

Although the linear (engineering) strains demonstrate a similar initial strain as inTL and UL, its variation and magnitude do not follow the same course. The bi-axial shear strains G−L display almost matching results forTL’s Green-Lagrange strain Exy andUL’s Almansi A G−L strain measure Exy. Although obvious diﬀerences can be observed in variation of Exy A and Exy through deformation, bothTL andUL approach the same ﬁnal solution. Chapter 5. Numerical results in plane strain 118

The axial normal stresses demonstrate greater stress generated by Cauchy and Kirchhoﬀ stress, i.e. Txx and Tyy than second Piola-Kirchhoﬀ and engineering stresses, i.e. Syy and

σyy. In the same manner the transversal stresses associated withUL display a greater magnitude than stresses inTL. However the shear strains associated withTL andUL appear to follow the same path and magnitude with slightly greater second Piola-Kirchhoff stress around time step 1 [s]. The linear engineering stresses follow a different path and magnitude in comparison with non-linear stresses. The stresses start with an initial stress value as the dynamic analysis has started at the final configuration when the cantilever has been deformed.

(a) (b) 5 10

0 5

-5 0

-10 -5 0 1 2.5 0 1 2.5 S T

(c)

Stress [MPa] 5

-5

-10 0 1 2.5 Time [s]

Figure 47 – (a) Axial (b) transversal and (c) shear stresses in plane strain (dynamic). 119

Chapter 6

Numerical & experimental results in 3D

umerical and experimental outcome of this study in analysis of wood under large deformation is presented in this chapter. In static analysis a Total Lagrangian N approach is considered and executed inside a Newton-Raphson scheme for 20 load steps. The wood board is numerically represented by a mesh of 5 [mm] constant strain tetrahedral FEM elements and clamped at one end and a vertical displacement of 200 [mm] is applied at its free end. The static results concern numerical and experimental investigation of horizontal displacement of the free end in relation to the applied vertical displacement. Further, numerical results associated with static internal and external quantities are presented. In rate-dependency (dynamic) analysis a Newmark-β time-integration method based on 1 1 TL with stability coefficients γ = 2 and β = 2 is proposed. The choice of mentioned stability coefficients introduce algorithmic damping to the system. The dynamic analysis is executed in a Newton-Raphson scheme for 8 [s] with a time-step length of ∆t = 1/1000 corresponding to 8000 time-steps. The experimental data is extracted from accelerometers placed on the wood board throughout the experiment, Figure 31. The signal data has been processed, filtered with Butterworth filer and integrated using cumulative trapezoidal numerical rule to determine corresponding accelerations, velocities and displacements. In the Butterworth filter a Nyquist–Shannon sampling theorem is utilized to determine discrete sequence of samples and extract finite information from the accelerometers signal, [22]. Chapter 6. Numerical & experimental results in 3D 120

6.1 Static results

The horizontal and vertical displacements of the wood board under a static deformation of 200 [mm] in Total Lagrangian versus experimental results demonstrate contraction of the wood board through ﬁnite deformation. In purpose of clear visualization the ﬁgure below is presented in a mesh of element size corresponding to 20 [mm]. However, the analysis is performed with element size of 5 [mm]

Figure 48 – Static deformation in of wood in Total Lagrangian with colormap of displacements.

The numerical and experimental horizontal and vertical displacements indicate a diﬀer- ence of approximately 10%. Figure 49 displays horizontal and vertical displacements of TL and experimental data with 10% error bars. The error appear to increase in magnitude as the imposed displacement increment increases, noting that the displacement is applied in 20 increments.

TL Experimental Data

200

180

160

140

120

100

Vertical displacement [mm] 40

0 0 2 4 6 8 10 12 14 16 18 20 Horizontal displacement [mm] Figure 49 – Static deformation inTL versus experimental data with 10% bars. Chapter 6. Numerical & experimental results in 3D 121

Despite the horizontal and vertical deformation, the board intends to displace in lateral direction in the numerical model due to grain direction of the wood board and numerical coordinate transformation. However, this has not been investigated experimentally due to lack of adequate equipment.

-50

-100

-150 V. displacement [mm]

-200 0 -5 0 -0.02 -10 -0.04 -15 -0.06 -0.08 H. displacement [mm] -20 -0.1 L. displacement [mm]

Figure 50 – Horizontal, vertical and lateral displacement of wood.

Further numerical results can be found in AppendixB. Chapter 6. Numerical & experimental results in 3D 122

6.2 Implicit dynamic results

Dynamic motion of the wood board in the numerical model at different configurations is presented below. It is observed that the dynamic analysis is executed at an initial vertical deformation of 200 [mm]. However, unlike the analysis in 2D where no damping was considered, a Newmark-Rayleigh damping coefficient of d1 = 0.77 is introduced to the board. It must be emphasized that in this work, the damping coefficient is only proportional to the material mass. Thus, when the board oscillates upwards the maximum displacement amplitude does not reach 200 [mm]. The dynamic results of the proposed model (Total LagrangianTL based Newmark- β method) in comparison with experimental data associated with each accelerometer in Figure 31 are presented.

Figure 51 – Dynamic motion of wood in the proposed numerical model.

In the modal analysis, the following eigenfrequencies are found, Table6. Thus, the band- pass is designed with respect to the lowest signals with a frequency of fs = 2.5 [Hz], Equation (382).

Table 6 – Modal eigenfrequency of wood.

Mode No. Frequency [Hz] 1 4.226 2 26.399 3 73.529

The numerical model appear to be able to predict dynamic behaviour of the wood board. However, under- and overshoot in the velocities and displacements are observed. Although the acceleration responses follow similar path, the numerical model appear to oscillate more than the experimental data in rising and diminishing. In the Figures 52, 53 and 54 it is observed that the displacements associated with the accelerometers data starts from zero, due to the fact that the accelerometers do not recognize the initial displacement. Thus, the initial displacement is disregarded. Chapter 6. Numerical & experimental results in 3D 123

Proposed Experimental data ] 2 105 1

-1

-2 0 2 4 6 8 Acceleration [mm/s

2000 0 -2000 -4000 Velocity [mm/s] 0 2 4 6 8

200

-200

Displacement [mm] 0 2 4 6 8 Time [s]

Figure 52 – Comparison of experimental data from accelerometer 1 and the corresponding numerical model.

Proposed Experimental data ] 2 104

-10

-20 0 2 4 6 8 Acceleration [mm/s

2000

Velocity [mm/s] -2000 0 2 4 6 8

100

-100

Displacement [mm] 0 2 4 6 8 Time [s]

Figure 53 – Comparison of experimental data from accelerometer 2 and the corresponding numerical model. Chapter 6. Numerical & experimental results in 3D 124

Although numerical model in comparison with the experimental data produce under- and overshoot, the numerical model is capable of predicting kinematics appropriately. The numerical model appear accelerate more than the recorded signal from the accelerometers. However the velocity with minor under- and overshoot matching the experimental data.

Proposed Experimental data ] 2 105 1

-1 0 2 4 6 8 Acceleration [mm/s

1000

Velocity [mm/s] -1000 0 2 4 6 8

-50

Displacement [mm] 0 2 4 6 8 Time [s]

Figure 54 – Comparison of experimental data from accelerometer 3 and the corresponding numerical model.

Further numerical solutions associated with internal force, stress and strain variation through dynamic motion can be found in AppendixB. 125

Chapter 7

Analysis & Conclusion

7.1 Analysis

The Updated Lagrangian formulationUL in this work was derived from Total La- grangian formulationTL. Consequently, identical solution were obtained with respect to final configuration kinematics. However,UL can be formulated purely in terms of Almansi strain and Cauchy stress. In this manner the the formulations produce minor deviation, whereUL is computationally more effective and expensive thanTL. However, with a fine mesh both formulations approach similar solution withUL being computationally more expensive, also investigated by [13]. In structural analysisTL may be appropriate to employ, while in Computational Fluid Dynamics and history dependent problems such as plasticity and creep analysis,UL may be more adequate. This conviction stems from the fact that inUL the boundary nodes do not coincide with material particles and kinematics at each configuration must be updated with the kinematics from a previous time-step. Accuracy in direct time integration methods are still questionable in structural dynamics. Thus, continuous research are seeking to address this issue in an appropriate manner. Newmark-β is among the best classical direct time-integration methods still in use, while other methods such as Wilson-θ is no longer used. However, modern methods such as

Bathe ρ∞ has proven to be more accurate than Newmark-β. This method has been explored, however not implemented in this work due to time constrains. Dynamic analysis of wood using direct time integration methods is still rare. Hence, further research may consider Bathe’s ρ∞-time integration method with controllable spectral radius [7], to archive a better prediction of dynamic behaviour of wood. Due to limited resources and appropriate equipment, uncertainties in the experimental data is probable. Thus, the full potential of Newmark-β time-integration method may not have been utilized. Uncertainties in the material properties of wood is highly likely due to the fact that wood is a porous and organic material with a natural composite of cellulose fibers. In this work oregon pine properties proposed by [18] are utilized. However, a wide range of different data is available. Thus, further research may consider stochastic based data to minimize Chapter 7. Analysis & Conclusion 126 uncertainties. In the present work, a damping solution is also determined for wood, which may open the door for further investigation considering contribution from material stiffness in addition to mass. The damping coefficient can be utilized in sorting timber in different classes according to dynamic properties. The dynamic properties of wood are significant in building tall and durable structures. For example, wood based transmission towers and multi-storey buildings. This work has proven the fact that wood is capable of undergoing large static and dynamic deformations, which linear engineering solution fails to predict. Thus, the use of continuum mechanics based approaches appear to be inevitable in predictive modeling of long lasting and sustainable structures. Chapter 7. Analysis & Conclusion 127

7.2 Conclusion

Lagrangian and Eulerian descriptions, i.e. Total LagrangianTL and Updated La- grangianUL are undoubtedly capable of predicting material behaviour appropriately. However, the engineering analysis has been inadequate in analysis and insuﬃcient in predicting material kinematics under large static and dynamic deformations. Although uncertainties in direct time-integration methods have been recognized, the Newmark-β has been to a certain extend able to predict dynamic behaviour of orthotropic wood under large deformation. The numerical model can be further developed to become operational in sorting timber products according to dynamic properties in the wood industry. 128

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Appendix 131

APPENDIX A

Numerical functions

A.1 Green-Lagrange strain and deformation gradient

Purpose: Compute Green-Lagrange strain and deformation gradient for a 3-node triangular element under large deformation in plane strain.

a4 (x3, y3) a5

a 2 a3 ey (x2, y2) a1

(x1, y1)

ex Three-node triangular FEM element

[ee, eﬀ] = plan_gs(ex, ey, ed)

Description: plan_gs computes element Green-Lagrange strain ee, and deformation gradient eff, for a 3-node triangular FEM element under large deformation. Input: Undeformed element coordinates ex, ey and displacement vector ed in the current configuration. h i ex = x1 x2 x3 h i ey = y1 y2 y3 h i ed = a1 a2 a3 a4 a5 a6 Output: Green-Lagrange strain vector ee and deformation gradient eff. h iT ee = E11 E22 E12 h iT eff = ∂x1/∂X1 ∂x1/∂X2 ∂x2/∂X1 ∂x2/∂X2 APPENDIX A. Numerical functions 132

A.2 Internal forces

Purpose: Compute internal force vector for a 3-node triangular element under large deformation in plane strain.

a4 (x3, y3) a5

a 2 a3

ey (x2, y2) a1

(x1, y1)

Three-node triangular FEM element

[ef] = plan_gf(ex, ey, ed, es, t)

Description: plan_gs computes element internal force vector for a 3-node triangular FEM element under large deformation. Input: Undeformed element coordinates ex, ey, displacement vector ed in the current conﬁguration, element 2nd Piola-Kirchhoﬀ stress es and thickness of the element t. h i ex = x1 x2 x3 h i ey = y1 y2 y3 h i ed = a1 a2 a3 a4 a5 a6 h iT es = S11 Syy S12

Output: Internal force vector ef.

T h e e e e e e i ef = f1 f2 f3 f4 f5 f6 APPENDIX A. Numerical functions 133

A.3 Tangent stiﬀness matrix

Purpose: Compute tangent stiﬀness matrix vector for a 3-node triangular element under large deformation in plane strain.

a4 (x3, y3) a5

a 2 a3

ey (x2, y2) a1

(x1, y1)

Three-node triangular FEM element

[Ke] = stiﬀ_ge(ex, ey, D, ed, es, t)

Description: stiff_ge computes element tangent stiffness matrix for a 3-node triangular FEM element under large deformation. Input: Undeformed element coordinates ex, ey, element constitutive matrix D, displacement vector ed in the current configuration, element 2nd Piola-Kirchhoff stress es and thickness of the element t. h i ex = x1 x2 x3 h i ey = y1 y2 y3   D11 D12 D13     D = D21 D22 D23   D31 D32 D33 h i ed = a1 a2 a3 a4 a5 a6 h iT es = S11 Syy S12

Output: Tangent stiﬀness matrix Ke.   k11 . . . k16    . . .  Ke =  . .. .    k61 . . . k66 134

APPENDIX B

Additional numerical results in 3D

B.1 Static results

Reaction force Internal force

10-8 (a) 0 -5 -10 -15

-16 -14 -12 -10 -8 -6 -4 -2

(b) 150 100 50 Force [N] -200 -180 -160 -140 -120 -100 -80 -60 -40 -20

10-10 (c) 15 10 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Displacement [mm]

Figure 55 – Static (a) Horizontal, (b) vertical and (c) lateral internal and reaction forces and corresponding displacements of wood. APPENDIX B. Additional numerical results in 3D 135

EG-L EG-L EG-L 10-6 xx 10-7 yy 10-7 zz

6 -2 -1 5 4 -4 -2 3 2 -6 -3 1 -8 -4 10 20 10 20 10 20 EG-L EG-L EG-L 10-5 xy 10-6 yz 10-6 xz Strain [-] -1 -1 -2 -2 -2 -4 -3 -3 -4 -6 -4 -5 -6 -8 -5 10 20 10 20 10 20 Load-step [-]

Figure 56 – Strains in wood under static loading in 3D.

S S S xx 10-3 yy 10-3 zz

1 -1 -0.5 0.8 -2 -1 0.6 -3 -4 -1.5 0.4 -5 -2 0.2 -6 -2.5 10 20 10 20 10 20 S S S xy yz 10-3 xz

Stress [MPa] -0.2 -0.01 -1

-2 -0.4 -0.02 -3 -0.6 -0.03 -4 -0.8 -0.04 -5 10 20 10 20 10 20 Load-step [-]

Figure 57 – Stresses in wood under static loading in 3D. APPENDIX B. Additional numerical results in 3D 136

B.2 Dynamic results

(a)

0.4 0.2 0 -0.2 -16 -14 -12 -10 -8 -6 -4 -2 0

(b)

2 1 0

Force [N] -1

-150 -100 -50 0 50 100 150

-0.4 -0.2 0 0.2 0.4 0.6 Displacement [mm]

Figure 58 – Dynamic (a) Horizontal, (b) vertical and (c) lateral internal forces and corresponding displacements of wood.

EG-L EG-L EG-L 10-7 xx 10-7 yy 10-7 zz 10 6 2 4 5 1 2 0 0 0 -1 -2

-4 -5 -2 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 EG-L EG-L EG-L 10-6 xy 10-6 yz 10-6 xz

Strain [-] 4 1 1

2 0.5 0.5 0 0 0 -2 -0.5 -4 -0.5 -1 -6 -1 -1.5 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 Time [s]

Figure 59 – Strains in wood under dynamic loading. APPENDIX B. Additional numerical results in 3D 137

S S S xx 10-3 yy 10-3 zz 0.15 10 4

0.1 5 2 0.05

0 0 0 -0.05

-0.1 -5 -2 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 S S S xy yz 10-3 xz 0.05 0.01 1 Stress [MPa] 0.005 0.5 0

0 0

-0.05 -0.005 -0.5

-0.1 -0.01 -1 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 Time [s]

Figure 60 – Stresses in wood under dynamic loading. Lnu.se Faculty of Technology 351 95 Växjö, Sweden